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Properties

Label 539.2.e.m
Level $539$
Weight $2$
Character orbit 539.e
Analytic conductor $4.304$
Analytic rank $0$
Dimension $6$
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] = [539,2,Mod(67,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.e (of order \(3\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{2}+ \cdots + (\beta_{5} - \beta_{4} + \cdots + 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{2}+ \cdots + (\beta_{4} - 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 6 q^{5} + 6 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 6 q^{5} + 6 q^{6} + 6 q^{8} + 3 q^{10} - 3 q^{11} + 3 q^{12} - 6 q^{13} + 18 q^{15} + 6 q^{16} - 3 q^{17} + 12 q^{18} + 9 q^{19} - 12 q^{20} + 6 q^{24} - 3 q^{25} + 9 q^{26} - 12 q^{27} - 6 q^{29} + 6 q^{30} + 9 q^{31} + 9 q^{32} + 3 q^{33} + 12 q^{34} + 6 q^{36} - 3 q^{39} + 3 q^{40} - 18 q^{41} + 3 q^{45} + 24 q^{46} - 3 q^{47} + 36 q^{48} + 30 q^{50} + 18 q^{51} + 9 q^{52} - 9 q^{53} - 18 q^{54} - 12 q^{55} - 15 q^{58} - 6 q^{60} + 12 q^{61} + 6 q^{62} - 6 q^{64} - 15 q^{65} - 3 q^{66} - 21 q^{68} - 42 q^{69} - 18 q^{71} - 12 q^{72} + 6 q^{73} - 18 q^{74} + 9 q^{75} + 18 q^{76} + 12 q^{78} + 3 q^{79} - 27 q^{80} - 15 q^{81} - 3 q^{82} + 30 q^{83} - 54 q^{85} + 9 q^{86} - 24 q^{87} - 3 q^{88} - 15 q^{89} + 72 q^{90} - 6 q^{92} - 6 q^{93} - 9 q^{95} + 3 q^{96} - 90 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display

Display \(\zeta_{18}^j\) in terms of \(\beta_i\)

\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{18}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 − 0.984808i
−0.766044 − 0.642788i
0.939693 − 0.342020i
−0.173648 + 0.984808i
−0.939693 − 1.62760i 0.326352 − 0.565258i −0.766044 + 1.32683i 1.76604 + 3.05888i −1.22668 0 −0.879385 1.28699 + 2.22913i 3.31908 − 5.74881i
67.2 0.173648 + 0.300767i −0.266044 + 0.460802i 0.939693 − 1.62760i 0.0603074 + 0.104455i −0.184793 0 1.34730 1.35844 + 2.35289i −0.0209445 + 0.0362770i
67.3 0.766044 + 1.32683i 1.43969 − 2.49362i −0.173648 + 0.300767i 1.17365 + 2.03282i 4.41147 0 2.53209 −2.64543 − 4.58202i −1.79813 + 3.11446i
177.1 −0.939693 + 1.62760i 0.326352 + 0.565258i −0.766044 − 1.32683i 1.76604 − 3.05888i −1.22668 0 −0.879385 1.28699 − 2.22913i 3.31908 + 5.74881i
177.2 0.173648 − 0.300767i −0.266044 − 0.460802i 0.939693 + 1.62760i 0.0603074 − 0.104455i −0.184793 0 1.34730 1.35844 − 2.35289i −0.0209445 − 0.0362770i
177.3 0.766044 − 1.32683i 1.43969 + 2.49362i −0.173648 − 0.300767i 1.17365 − 2.03282i 4.41147 0 2.53209 −2.64543 + 4.58202i −1.79813 − 3.11446i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.e.m 6
7.b odd 2 1 77.2.e.a ✓ 6
7.c even 3 1 539.2.a.g 3
7.c even 3 1 inner 539.2.e.m 6
7.d odd 6 1 77.2.e.a ✓ 6
7.d odd 6 1 539.2.a.j 3
21.c even 2 1 693.2.i.h 6
21.g even 6 1 693.2.i.h 6
21.g even 6 1 4851.2.a.bj 3
21.h odd 6 1 4851.2.a.bk 3
28.d even 2 1 1232.2.q.m 6
28.f even 6 1 1232.2.q.m 6
28.f even 6 1 8624.2.a.ch 3
28.g odd 6 1 8624.2.a.co 3
77.b even 2 1 847.2.e.c 6
77.h odd 6 1 5929.2.a.u 3
77.i even 6 1 847.2.e.c 6
77.i even 6 1 5929.2.a.x 3
77.j odd 10 4 847.2.n.g 24
77.l even 10 4 847.2.n.f 24
77.n even 30 4 847.2.n.f 24
77.p odd 30 4 847.2.n.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.a ✓ 6 7.b odd 2 1
77.2.e.a ✓ 6 7.d odd 6 1
539.2.a.g 3 7.c even 3 1
539.2.a.j 3 7.d odd 6 1
539.2.e.m 6 1.a even 1 1 trivial
539.2.e.m 6 7.c even 3 1 inner
693.2.i.h 6 21.c even 2 1
693.2.i.h 6 21.g even 6 1
847.2.e.c 6 77.b even 2 1
847.2.e.c 6 77.i even 6 1
847.2.n.f 24 77.l even 10 4
847.2.n.f 24 77.n even 30 4
847.2.n.g 24 77.j odd 10 4
847.2.n.g 24 77.p odd 30 4
1232.2.q.m 6 28.d even 2 1
1232.2.q.m 6 28.f even 6 1
4851.2.a.bj 3 21.g even 6 1
4851.2.a.bk 3 21.h odd 6 1
5929.2.a.u 3 77.h odd 6 1
5929.2.a.x 3 77.i even 6 1
8624.2.a.ch 3 28.f even 6 1
8624.2.a.co 3 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{6} + 3T_{2}^{4} - 2T_{2}^{3} + 9T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 2T_{3}^{3} + 3T_{3}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 6 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{6} + 57 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} - 36 T + 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{6} + 36 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T + 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + \cdots + 104329 \) Copy content Toggle raw display
$53$ \( T^{6} + 9 T^{5} + \cdots + 210681 \) Copy content Toggle raw display
$59$ \( T^{6} + 93 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} + \cdots + 576 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{3} + 9 T^{2} + \cdots - 801)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} - 3 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$83$ \( (T^{3} - 15 T^{2} + \cdots + 267)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 15 T^{5} + \cdots + 12321 \) Copy content Toggle raw display
$97$ \( (T^{3} + 45 T^{2} + \cdots + 3329)^{2} \) Copy content Toggle raw display
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