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Properties

Label 539.2.e.n
Level $539$
Weight $2$
Character orbit 539.e
Analytic conductor $4.304$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.6927565824.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 23x^{4} + 10x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{2} + \beta_{2} q^{3} + (\beta_{5} + 3 \beta_{4}) q^{4} + (\beta_{7} - \beta_{6} + \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{6} + \beta_1) q^{6} + ( - \beta_{3} + 5) q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{2} + \beta_{2} q^{3} + (\beta_{5} + 3 \beta_{4}) q^{4} + (\beta_{7} - \beta_{6} + \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{6} + \beta_1) q^{6} + ( - \beta_{3} + 5) q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{9} - 4 \beta_{2} q^{10} - \beta_{4} q^{11} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{2}) q^{12} + (2 \beta_{7} + \beta_{6} - \beta_1) q^{13} + (3 \beta_{3} - 4) q^{15} + ( - 3 \beta_{5} - 3 \beta_{4} + \cdots - 3) q^{16}+ \cdots + (\beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 10 q^{4} + 36 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 10 q^{4} + 36 q^{8} - 2 q^{9} + 4 q^{11} - 20 q^{15} - 6 q^{16} - 18 q^{18} - 4 q^{22} + 14 q^{23} - 26 q^{25} + 16 q^{29} + 56 q^{30} - 18 q^{32} + 44 q^{36} - 26 q^{37} - 32 q^{39} - 16 q^{43} + 10 q^{44} + 24 q^{46} + 60 q^{50} - 4 q^{51} + 8 q^{53} - 56 q^{57} - 4 q^{58} + 76 q^{60} + 28 q^{64} - 16 q^{65} - 6 q^{67} - 52 q^{71} - 26 q^{72} + 4 q^{74} + 32 q^{78} - 36 q^{79} + 24 q^{81} + 72 q^{85} + 72 q^{86} + 18 q^{88} - 104 q^{92} - 6 q^{93} - 20 q^{95} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 23x^{4} + 10x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 82\nu ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 36 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 23\nu^{4} - 115\nu^{2} - 50 ) / 46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 23\nu^{4} - 92\nu^{2} - 40 ) / 23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 23\nu^{5} + 115\nu^{3} + 50\nu ) / 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10\nu^{7} - 46\nu^{5} - 207\nu^{3} - 8\nu ) / 23 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} - \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 8\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} - 9\beta_{6} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{3} + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{2} + 41\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\)

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
0.331077 − 0.573442i
−0.331077 + 0.573442i
−1.06789 + 1.84964i
1.06789 − 1.84964i
0.331077 + 0.573442i
−0.331077 − 0.573442i
−1.06789 − 1.84964i
1.06789 + 1.84964i
−1.28078 − 2.21837i −1.17915 + 2.04234i −2.28078 + 3.95042i 1.84130 + 3.18923i 6.04090 0 6.56155 −1.28078 − 2.21837i 4.71659 − 8.16937i
67.2 −1.28078 − 2.21837i 1.17915 − 2.04234i −2.28078 + 3.95042i −1.84130 − 3.18923i −6.04090 0 6.56155 −1.28078 − 2.21837i −4.71659 + 8.16937i
67.3 0.780776 + 1.35234i −0.599676 + 1.03867i −0.219224 + 0.379706i −1.53610 − 2.66061i −1.87285 0 2.43845 0.780776 + 1.35234i 2.39871 − 4.15468i
67.4 0.780776 + 1.35234i 0.599676 − 1.03867i −0.219224 + 0.379706i 1.53610 + 2.66061i 1.87285 0 2.43845 0.780776 + 1.35234i −2.39871 + 4.15468i
177.1 −1.28078 + 2.21837i −1.17915 − 2.04234i −2.28078 − 3.95042i 1.84130 − 3.18923i 6.04090 0 6.56155 −1.28078 + 2.21837i 4.71659 + 8.16937i
177.2 −1.28078 + 2.21837i 1.17915 + 2.04234i −2.28078 − 3.95042i −1.84130 + 3.18923i −6.04090 0 6.56155 −1.28078 + 2.21837i −4.71659 − 8.16937i
177.3 0.780776 − 1.35234i −0.599676 − 1.03867i −0.219224 − 0.379706i −1.53610 + 2.66061i −1.87285 0 2.43845 0.780776 − 1.35234i 2.39871 + 4.15468i
177.4 0.780776 − 1.35234i 0.599676 + 1.03867i −0.219224 − 0.379706i 1.53610 − 2.66061i 1.87285 0 2.43845 0.780776 − 1.35234i −2.39871 − 4.15468i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.e.n 8
7.b odd 2 1 inner 539.2.e.n 8
7.c even 3 1 539.2.a.k ✓ 4
7.c even 3 1 inner 539.2.e.n 8
7.d odd 6 1 539.2.a.k ✓ 4
7.d odd 6 1 inner 539.2.e.n 8
21.g even 6 1 4851.2.a.bv 4
21.h odd 6 1 4851.2.a.bv 4
28.f even 6 1 8624.2.a.cu 4
28.g odd 6 1 8624.2.a.cu 4
77.h odd 6 1 5929.2.a.ba 4
77.i even 6 1 5929.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.a.k ✓ 4 7.c even 3 1
539.2.a.k ✓ 4 7.d odd 6 1
539.2.e.n 8 1.a even 1 1 trivial
539.2.e.n 8 7.b odd 2 1 inner
539.2.e.n 8 7.c even 3 1 inner
539.2.e.n 8 7.d odd 6 1 inner
4851.2.a.bv 4 21.g even 6 1
4851.2.a.bv 4 21.h odd 6 1
5929.2.a.ba 4 77.h odd 6 1
5929.2.a.ba 4 77.i even 6 1
8624.2.a.cu 4 28.f even 6 1
8624.2.a.cu 4 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 \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{8} + 7T_{3}^{6} + 41T_{3}^{4} + 56T_{3}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + 5 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 7 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{8} + 23 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 56 T^{2} + 512)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 20 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{8} + 28 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$23$ \( (T^{4} - 7 T^{3} + 41 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 79 T^{6} + \cdots + 1827904 \) Copy content Toggle raw display
$37$ \( (T^{4} + 13 T^{3} + \cdots + 1444)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 64)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 20 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + 175 T^{6} + \cdots + 25000000 \) Copy content Toggle raw display
$61$ \( T^{8} + 244 T^{6} + \cdots + 133448704 \) Copy content Toggle raw display
$67$ \( (T^{4} + 3 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 13 T - 64)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 180 T^{6} + \cdots + 6718464 \) Copy content Toggle raw display
$79$ \( (T^{4} + 18 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 80 T^{2} + 512)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 1368408064 \) Copy content Toggle raw display
$97$ \( (T^{4} - 335 T^{2} + 5408)^{2} \) Copy content Toggle raw display
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