Properties

Label 4003.2.a.a
Level $4003$
Weight $2$
Character orbit 4003.a
Self dual yes
Analytic conductor $31.964$
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] = [4003,2,Mod(1,4003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4003, 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("4003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4003.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: \(31.9641159291\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + \beta q^{3} + \beta q^{5} + 2 q^{6} - q^{7} - 2 \beta q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + \beta q^{3} + \beta q^{5} + 2 q^{6} - q^{7} - 2 \beta q^{8} - q^{9} + 2 q^{10} + ( - 2 \beta - 2) q^{11} + (2 \beta + 4) q^{13} - \beta q^{14} + 2 q^{15} - 4 q^{16} + ( - 4 \beta - 1) q^{17} - \beta q^{18} + (4 \beta + 1) q^{19} - \beta q^{21} + ( - 2 \beta - 4) q^{22} + (2 \beta - 6) q^{23} - 4 q^{24} - 3 q^{25} + (4 \beta + 4) q^{26} - 4 \beta q^{27} + ( - 2 \beta - 4) q^{29} + 2 \beta q^{30} + ( - 6 \beta + 2) q^{31} + ( - 2 \beta - 4) q^{33} + ( - \beta - 8) q^{34} - \beta q^{35} + ( - 2 \beta - 7) q^{37} + (\beta + 8) q^{38} + (4 \beta + 4) q^{39} - 4 q^{40} + (2 \beta + 8) q^{41} - 2 q^{42} - \beta q^{45} + ( - 6 \beta + 4) q^{46} + (5 \beta + 4) q^{47} - 4 \beta q^{48} - 6 q^{49} - 3 \beta q^{50} + ( - \beta - 8) q^{51} + ( - 8 \beta - 2) q^{53} - 8 q^{54} + ( - 2 \beta - 4) q^{55} + 2 \beta q^{56} + (\beta + 8) q^{57} + ( - 4 \beta - 4) q^{58} + (2 \beta - 6) q^{59} + (2 \beta + 10) q^{61} + (2 \beta - 12) q^{62} + q^{63} + 8 q^{64} + (4 \beta + 4) q^{65} + ( - 4 \beta - 4) q^{66} + ( - 4 \beta + 7) q^{67} + ( - 6 \beta + 4) q^{69} - 2 q^{70} + 6 q^{71} + 2 \beta q^{72} - 3 q^{73} + ( - 7 \beta - 4) q^{74} - 3 \beta q^{75} + (2 \beta + 2) q^{77} + (4 \beta + 8) q^{78} + ( - 6 \beta + 3) q^{79} - 4 \beta q^{80} - 5 q^{81} + (8 \beta + 4) q^{82} + ( - 10 \beta + 3) q^{83} + ( - \beta - 8) q^{85} + ( - 4 \beta - 4) q^{87} + (4 \beta + 8) q^{88} + (2 \beta - 9) q^{89} - 2 q^{90} + ( - 2 \beta - 4) q^{91} + (2 \beta - 12) q^{93} + (4 \beta + 10) q^{94} + (\beta + 8) q^{95} + (3 \beta + 6) q^{97} - 6 \beta q^{98} + (2 \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{6} - 2 q^{7} - 2 q^{9} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 2 q^{19} - 8 q^{22} - 12 q^{23} - 8 q^{24} - 6 q^{25} + 8 q^{26} - 8 q^{29} + 4 q^{31} - 8 q^{33} - 16 q^{34} - 14 q^{37} + 16 q^{38} + 8 q^{39} - 8 q^{40} + 16 q^{41} - 4 q^{42} + 8 q^{46} + 8 q^{47} - 12 q^{49} - 16 q^{51} - 4 q^{53} - 16 q^{54} - 8 q^{55} + 16 q^{57} - 8 q^{58} - 12 q^{59} + 20 q^{61} - 24 q^{62} + 2 q^{63} + 16 q^{64} + 8 q^{65} - 8 q^{66} + 14 q^{67} + 8 q^{69} - 4 q^{70} + 12 q^{71} - 6 q^{73} - 8 q^{74} + 4 q^{77} + 16 q^{78} + 6 q^{79} - 10 q^{81} + 8 q^{82} + 6 q^{83} - 16 q^{85} - 8 q^{87} + 16 q^{88} - 18 q^{89} - 4 q^{90} - 8 q^{91} - 24 q^{93} + 20 q^{94} + 16 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−1.41421 −1.41421 0 −1.41421 2.00000 −1.00000 2.82843 −1.00000 2.00000
1.2 1.41421 1.41421 0 1.41421 2.00000 −1.00000 −2.82843 −1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4003\) \( +1 \)

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 4003.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4003.2.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4003))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2 \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 31 \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$37$ \( T^{2} + 14T + 41 \) Copy content Toggle raw display
$41$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$61$ \( T^{2} - 20T + 92 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 17 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( (T + 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 63 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 191 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 73 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
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