Properties

Label 538.2.a.b
Level $538$
Weight $2$
Character orbit 538.a
Self dual yes
Analytic conductor $4.296$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + ( - \beta + 1) q^{5} + \beta q^{6} - q^{7} + q^{8} + \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} + ( - \beta + 1) q^{5} + \beta q^{6} - q^{7} + q^{8} + \beta q^{9} + ( - \beta + 1) q^{10} + 3 q^{11} + \beta q^{12} + 5 q^{13} - q^{14} - 3 q^{15} + q^{16} + ( - 2 \beta - 1) q^{17} + \beta q^{18} + 2 q^{19} + ( - \beta + 1) q^{20} - \beta q^{21} + 3 q^{22} + (4 \beta - 1) q^{23} + \beta q^{24} + ( - \beta - 1) q^{25} + 5 q^{26} + ( - 2 \beta + 3) q^{27} - q^{28} + ( - \beta + 7) q^{29} - 3 q^{30} + ( - 2 \beta - 5) q^{31} + q^{32} + 3 \beta q^{33} + ( - 2 \beta - 1) q^{34} + (\beta - 1) q^{35} + \beta q^{36} + ( - 2 \beta + 1) q^{37} + 2 q^{38} + 5 \beta q^{39} + ( - \beta + 1) q^{40} + ( - 2 \beta - 4) q^{41} - \beta q^{42} + ( - 4 \beta - 3) q^{43} + 3 q^{44} - 3 q^{45} + (4 \beta - 1) q^{46} + ( - 3 \beta - 3) q^{47} + \beta q^{48} - 6 q^{49} + ( - \beta - 1) q^{50} + ( - 3 \beta - 6) q^{51} + 5 q^{52} + ( - 2 \beta + 3) q^{54} + ( - 3 \beta + 3) q^{55} - q^{56} + 2 \beta q^{57} + ( - \beta + 7) q^{58} + (5 \beta - 5) q^{59} - 3 q^{60} - q^{61} + ( - 2 \beta - 5) q^{62} - \beta q^{63} + q^{64} + ( - 5 \beta + 5) q^{65} + 3 \beta q^{66} + (4 \beta + 7) q^{67} + ( - 2 \beta - 1) q^{68} + (3 \beta + 12) q^{69} + (\beta - 1) q^{70} + (6 \beta - 6) q^{71} + \beta q^{72} + ( - 2 \beta + 1) q^{73} + ( - 2 \beta + 1) q^{74} + ( - 2 \beta - 3) q^{75} + 2 q^{76} - 3 q^{77} + 5 \beta q^{78} + (\beta - 5) q^{79} + ( - \beta + 1) q^{80} + ( - 2 \beta - 6) q^{81} + ( - 2 \beta - 4) q^{82} + (2 \beta + 4) q^{83} - \beta q^{84} + (\beta + 5) q^{85} + ( - 4 \beta - 3) q^{86} + (6 \beta - 3) q^{87} + 3 q^{88} + ( - \beta - 5) q^{89} - 3 q^{90} - 5 q^{91} + (4 \beta - 1) q^{92} + ( - 7 \beta - 6) q^{93} + ( - 3 \beta - 3) q^{94} + ( - 2 \beta + 2) q^{95} + \beta q^{96} + 2 \beta q^{97} - 6 q^{98} + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{8} + q^{9} + q^{10} + 6 q^{11} + q^{12} + 10 q^{13} - 2 q^{14} - 6 q^{15} + 2 q^{16} - 4 q^{17} + q^{18} + 4 q^{19} + q^{20} - q^{21} + 6 q^{22} + 2 q^{23} + q^{24} - 3 q^{25} + 10 q^{26} + 4 q^{27} - 2 q^{28} + 13 q^{29} - 6 q^{30} - 12 q^{31} + 2 q^{32} + 3 q^{33} - 4 q^{34} - q^{35} + q^{36} + 4 q^{38} + 5 q^{39} + q^{40} - 10 q^{41} - q^{42} - 10 q^{43} + 6 q^{44} - 6 q^{45} + 2 q^{46} - 9 q^{47} + q^{48} - 12 q^{49} - 3 q^{50} - 15 q^{51} + 10 q^{52} + 4 q^{54} + 3 q^{55} - 2 q^{56} + 2 q^{57} + 13 q^{58} - 5 q^{59} - 6 q^{60} - 2 q^{61} - 12 q^{62} - q^{63} + 2 q^{64} + 5 q^{65} + 3 q^{66} + 18 q^{67} - 4 q^{68} + 27 q^{69} - q^{70} - 6 q^{71} + q^{72} - 8 q^{75} + 4 q^{76} - 6 q^{77} + 5 q^{78} - 9 q^{79} + q^{80} - 14 q^{81} - 10 q^{82} + 10 q^{83} - q^{84} + 11 q^{85} - 10 q^{86} + 6 q^{88} - 11 q^{89} - 6 q^{90} - 10 q^{91} + 2 q^{92} - 19 q^{93} - 9 q^{94} + 2 q^{95} + q^{96} + 2 q^{97} - 12 q^{98} + 3 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
1.00000 −1.30278 1.00000 2.30278 −1.30278 −1.00000 1.00000 −1.30278 2.30278
1.2 1.00000 2.30278 1.00000 −1.30278 2.30278 −1.00000 1.00000 2.30278 −1.30278
\(n\): 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.b 2
3.b odd 2 1 4842.2.a.d 2
4.b odd 2 1 4304.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.a.b 2 1.a even 1 1 trivial
4304.2.a.c 2 4.b odd 2 1
4842.2.a.d 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 3 \) 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
$3$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 51 \) Copy content Toggle raw display
$29$ \( T^{2} - 13T + 39 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 23 \) Copy content Toggle raw display
$37$ \( T^{2} - 13 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T - 27 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 75 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 29 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 108 \) Copy content Toggle raw display
$73$ \( T^{2} - 13 \) Copy content Toggle raw display
$79$ \( T^{2} + 9T + 17 \) Copy content Toggle raw display
$83$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 11T + 27 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
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