Properties

Label 538.2.a.a
Level $538$
Weight $2$
Character orbit 538.a
Self dual yes
Analytic conductor $4.296$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta q^{3} + q^{4} + (\beta - 3) q^{5} - \beta q^{6} + (2 \beta - 3) q^{7} + q^{8} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta q^{3} + q^{4} + (\beta - 3) q^{5} - \beta q^{6} + (2 \beta - 3) q^{7} + q^{8} + (\beta - 2) q^{9} + (\beta - 3) q^{10} + ( - 2 \beta - 1) q^{11} - \beta q^{12} - q^{13} + (2 \beta - 3) q^{14} + (2 \beta - 1) q^{15} + q^{16} + ( - 2 \beta - 1) q^{17} + (\beta - 2) q^{18} + 2 q^{19} + (\beta - 3) q^{20} + (\beta - 2) q^{21} + ( - 2 \beta - 1) q^{22} + (2 \beta - 7) q^{23} - \beta q^{24} + ( - 5 \beta + 5) q^{25} - q^{26} + (4 \beta - 1) q^{27} + (2 \beta - 3) q^{28} + ( - 3 \beta - 1) q^{29} + (2 \beta - 1) q^{30} + ( - 4 \beta + 5) q^{31} + q^{32} + (3 \beta + 2) q^{33} + ( - 2 \beta - 1) q^{34} + ( - 7 \beta + 11) q^{35} + (\beta - 2) q^{36} + (6 \beta - 5) q^{37} + 2 q^{38} + \beta q^{39} + (\beta - 3) q^{40} + ( - 6 \beta + 4) q^{41} + (\beta - 2) q^{42} + ( - 2 \beta + 1) q^{43} + ( - 2 \beta - 1) q^{44} + ( - 4 \beta + 7) q^{45} + (2 \beta - 7) q^{46} + (3 \beta - 5) q^{47} - \beta q^{48} + ( - 8 \beta + 6) q^{49} + ( - 5 \beta + 5) q^{50} + (3 \beta + 2) q^{51} - q^{52} + (4 \beta - 1) q^{54} + (3 \beta + 1) q^{55} + (2 \beta - 3) q^{56} - 2 \beta q^{57} + ( - 3 \beta - 1) q^{58} + (9 \beta - 1) q^{59} + (2 \beta - 1) q^{60} + (4 \beta - 3) q^{61} + ( - 4 \beta + 5) q^{62} + ( - 5 \beta + 8) q^{63} + q^{64} + ( - \beta + 3) q^{65} + (3 \beta + 2) q^{66} + (2 \beta + 3) q^{67} + ( - 2 \beta - 1) q^{68} + (5 \beta - 2) q^{69} + ( - 7 \beta + 11) q^{70} + ( - 6 \beta + 6) q^{71} + (\beta - 2) q^{72} + (6 \beta - 7) q^{73} + (6 \beta - 5) q^{74} + 5 q^{75} + 2 q^{76} - q^{77} + \beta q^{78} + (11 \beta - 7) q^{79} + (\beta - 3) q^{80} + ( - 6 \beta + 2) q^{81} + ( - 6 \beta + 4) q^{82} + (2 \beta - 4) q^{83} + (\beta - 2) q^{84} + (3 \beta + 1) q^{85} + ( - 2 \beta + 1) q^{86} + (4 \beta + 3) q^{87} + ( - 2 \beta - 1) q^{88} + (11 \beta - 3) q^{89} + ( - 4 \beta + 7) q^{90} + ( - 2 \beta + 3) q^{91} + (2 \beta - 7) q^{92} + ( - \beta + 4) q^{93} + (3 \beta - 5) q^{94} + (2 \beta - 6) q^{95} - \beta q^{96} + ( - 14 \beta + 4) q^{97} + ( - 8 \beta + 6) q^{98} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9} - 5 q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 3 q^{18} + 4 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{22} - 12 q^{23} - q^{24} + 5 q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 5 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{33} - 4 q^{34} + 15 q^{35} - 3 q^{36} - 4 q^{37} + 4 q^{38} + q^{39} - 5 q^{40} + 2 q^{41} - 3 q^{42} - 4 q^{44} + 10 q^{45} - 12 q^{46} - 7 q^{47} - q^{48} + 4 q^{49} + 5 q^{50} + 7 q^{51} - 2 q^{52} + 2 q^{54} + 5 q^{55} - 4 q^{56} - 2 q^{57} - 5 q^{58} + 7 q^{59} - 2 q^{61} + 6 q^{62} + 11 q^{63} + 2 q^{64} + 5 q^{65} + 7 q^{66} + 8 q^{67} - 4 q^{68} + q^{69} + 15 q^{70} + 6 q^{71} - 3 q^{72} - 8 q^{73} - 4 q^{74} + 10 q^{75} + 4 q^{76} - 2 q^{77} + q^{78} - 3 q^{79} - 5 q^{80} - 2 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} + 5 q^{85} + 10 q^{87} - 4 q^{88} + 5 q^{89} + 10 q^{90} + 4 q^{91} - 12 q^{92} + 7 q^{93} - 7 q^{94} - 10 q^{95} - q^{96} - 6 q^{97} + 4 q^{98} + q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 −1.61803 1.00000 −1.38197 −1.61803 0.236068 1.00000 −0.381966 −1.38197
1.2 1.00000 0.618034 1.00000 −3.61803 0.618034 −4.23607 1.00000 −2.61803 −3.61803
\(n\): 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.a 2
3.b odd 2 1 4842.2.a.f 2
4.b odd 2 1 4304.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.a.a 2 1.a even 1 1 trivial
4304.2.a.d 2 4.b odd 2 1
4842.2.a.f 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(538))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$43$ \( T^{2} - 5 \) Copy content Toggle raw display
$47$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 7T - 89 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 11 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T - 149 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 5T - 145 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 236 \) Copy content Toggle raw display
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