Properties

Label 538.2.a.c
Level $538$
Weight $2$
Character orbit 538.a
Self dual yes
Analytic conductor $4.296$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.a (trivial)

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: yes
Analytic conductor: \(4.29595162874\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_1 - 1) q^{6} + \beta_{3} q^{7} - q^{8} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{5} + (\beta_1 - 1) q^{6} + \beta_{3} q^{7} - q^{8} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{10} + ( - \beta_{2} + 2) q^{11} + ( - \beta_1 + 1) q^{12} + ( - \beta_{3} - \beta_{2} + 1) q^{13} - \beta_{3} q^{14} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{15} + q^{16} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{17}+ \cdots + (\beta_{3} + \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9} - 5 q^{10} + 7 q^{11} + 3 q^{12} + 4 q^{13} + q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 3 q^{18} + 2 q^{19} + 5 q^{20} - 5 q^{21} - 7 q^{22} + 16 q^{23} - 3 q^{24} - q^{25} - 4 q^{26} + 6 q^{27} - q^{28} - 8 q^{30} - q^{31} - 4 q^{32} + q^{33} - 4 q^{34} + 3 q^{35} + 3 q^{36} - 2 q^{37} - 2 q^{38} + 3 q^{39} - 5 q^{40} - q^{41} + 5 q^{42} + 9 q^{43} + 7 q^{44} + 8 q^{45} - 16 q^{46} + 13 q^{47} + 3 q^{48} - 15 q^{49} + q^{50} + 3 q^{51} + 4 q^{52} + 28 q^{53} - 6 q^{54} + 13 q^{55} + q^{56} + 10 q^{57} + 19 q^{59} + 8 q^{60} - 20 q^{61} + q^{62} + 12 q^{63} + 4 q^{64} + 5 q^{65} - q^{66} + 15 q^{67} + 4 q^{68} - 5 q^{69} - 3 q^{70} + 15 q^{71} - 3 q^{72} - 7 q^{73} + 2 q^{74} + 12 q^{75} + 2 q^{76} - 6 q^{77} - 3 q^{78} - 17 q^{79} + 5 q^{80} + 4 q^{81} + q^{82} + 6 q^{83} - 5 q^{84} - 29 q^{85} - 9 q^{86} - 34 q^{87} - 7 q^{88} + 20 q^{89} - 8 q^{90} - 18 q^{91} + 16 q^{92} - 5 q^{93} - 13 q^{94} - 6 q^{95} - 3 q^{96} + 3 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 4\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 6\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 6\beta _1 + 3 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.90570
0.487928
−0.344151
−2.04948
−1.00000 −1.90570 1.00000 0.655849 1.90570 2.04948 −1.00000 0.631706 −0.655849
1.2 −1.00000 0.512072 1.00000 −1.04948 −0.512072 −2.90570 −1.00000 −2.73778 1.04948
1.3 −1.00000 1.34415 1.00000 3.90570 −1.34415 −0.487928 −1.00000 −1.19326 −3.90570
1.4 −1.00000 3.04948 1.00000 1.48793 −3.04948 0.344151 −1.00000 6.29934 −1.48793
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(269\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.2.a.c 4
3.b odd 2 1 4842.2.a.j 4
4.b odd 2 1 4304.2.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.a.c 4 1.a even 1 1 trivial
4304.2.a.e 4 4.b odd 2 1
4842.2.a.j 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 3T_{3}^{3} - 3T_{3}^{2} + 10T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(538))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 7 T^{3} + \cdots - 13 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 52 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 16 T^{3} + \cdots + 52 \) Copy content Toggle raw display
$29$ \( T^{4} - 51 T^{2} + \cdots + 17 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} + \cdots - 67 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots - 188 \) Copy content Toggle raw display
$41$ \( T^{4} + T^{3} + \cdots + 52 \) Copy content Toggle raw display
$43$ \( T^{4} - 9 T^{3} + \cdots + 1937 \) Copy content Toggle raw display
$47$ \( T^{4} - 13 T^{3} + \cdots - 1172 \) Copy content Toggle raw display
$53$ \( (T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 19 T^{3} + \cdots - 188 \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + \cdots + 268 \) Copy content Toggle raw display
$67$ \( T^{4} - 15 T^{3} + \cdots + 101 \) Copy content Toggle raw display
$71$ \( T^{4} - 15 T^{3} + \cdots - 2228 \) Copy content Toggle raw display
$73$ \( T^{4} + 7 T^{3} + \cdots + 6481 \) Copy content Toggle raw display
$79$ \( T^{4} + 17 T^{3} + \cdots + 68 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{4} - 20 T^{3} + \cdots - 191 \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} + \cdots + 27196 \) Copy content Toggle raw display
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