Properties

Label 503.2.a.c
Level $503$
Weight $2$
Character orbit 503.a
Self dual yes
Analytic conductor $4.016$
Analytic rank $0$
Dimension $1$
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] = [503,2,Mod(1,503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(503, 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("503.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.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: \(4.01647522167\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + 3 q^{3} - q^{4} - 2 q^{5} + 3 q^{6} + 3 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 3 q^{3} - q^{4} - 2 q^{5} + 3 q^{6} + 3 q^{7} - 3 q^{8} + 6 q^{9} - 2 q^{10} + 3 q^{11} - 3 q^{12} + 5 q^{13} + 3 q^{14} - 6 q^{15} - q^{16} - 8 q^{17} + 6 q^{18} + 4 q^{19} + 2 q^{20} + 9 q^{21} + 3 q^{22} - 5 q^{23} - 9 q^{24} - q^{25} + 5 q^{26} + 9 q^{27} - 3 q^{28} - 6 q^{30} - 2 q^{31} + 5 q^{32} + 9 q^{33} - 8 q^{34} - 6 q^{35} - 6 q^{36} + 4 q^{37} + 4 q^{38} + 15 q^{39} + 6 q^{40} - 10 q^{41} + 9 q^{42} - q^{43} - 3 q^{44} - 12 q^{45} - 5 q^{46} - 3 q^{47} - 3 q^{48} + 2 q^{49} - q^{50} - 24 q^{51} - 5 q^{52} - 12 q^{53} + 9 q^{54} - 6 q^{55} - 9 q^{56} + 12 q^{57} + 12 q^{59} + 6 q^{60} - 11 q^{61} - 2 q^{62} + 18 q^{63} + 7 q^{64} - 10 q^{65} + 9 q^{66} + 7 q^{67} + 8 q^{68} - 15 q^{69} - 6 q^{70} - 8 q^{71} - 18 q^{72} - 6 q^{73} + 4 q^{74} - 3 q^{75} - 4 q^{76} + 9 q^{77} + 15 q^{78} - 4 q^{79} + 2 q^{80} + 9 q^{81} - 10 q^{82} + 15 q^{83} - 9 q^{84} + 16 q^{85} - q^{86} - 9 q^{88} - 12 q^{90} + 15 q^{91} + 5 q^{92} - 6 q^{93} - 3 q^{94} - 8 q^{95} + 15 q^{96} - 6 q^{97} + 2 q^{98} + 18 q^{99}+O(q^{100}) \) 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
0
1.00000 3.00000 −1.00000 −2.00000 3.00000 3.00000 −3.00000 6.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(503\) \(-1\)

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 503.2.a.c 1
3.b odd 2 1 4527.2.a.d 1
4.b odd 2 1 8048.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.c 1 1.a even 1 1 trivial
4527.2.a.d 1 3.b odd 2 1
8048.2.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{3} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T - 5 \) Copy content Toggle raw display
$17$ \( T + 8 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 5 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T + 3 \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 11 \) Copy content Toggle raw display
$67$ \( T - 7 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 15 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 6 \) Copy content Toggle raw display
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