Properties

Label 4004.2.m.a
Level $4004$
Weight $2$
Character orbit 4004.m
Analytic conductor $31.972$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4004,2,Mod(2157,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4004.2157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.m (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} + i q^{5} + i q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + i q^{5} + i q^{7} + q^{9} - i q^{11} + (3 i + 2) q^{13} + 2 i q^{15} - 4 q^{17} + 5 i q^{19} + 2 i q^{21} - 9 q^{23} + 4 q^{25} - 4 q^{27} + 5 q^{29} + 7 i q^{31} - 2 i q^{33} - q^{35} + 2 i q^{37} + (6 i + 4) q^{39} - 6 i q^{41} - 11 q^{43} + i q^{45} - i q^{47} - q^{49} - 8 q^{51} - q^{53} + q^{55} + 10 i q^{57} + 4 i q^{59} - 6 q^{61} + i q^{63} + (2 i - 3) q^{65} + 10 i q^{67} - 18 q^{69} + 4 i q^{71} + 3 i q^{73} + 8 q^{75} + q^{77} + 11 q^{79} - 11 q^{81} - 9 i q^{83} - 4 i q^{85} + 10 q^{87} - 15 i q^{89} + (2 i - 3) q^{91} + 14 i q^{93} - 5 q^{95} + 7 i q^{97} - i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 2 q^{9} + 4 q^{13} - 8 q^{17} - 18 q^{23} + 8 q^{25} - 8 q^{27} + 10 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 2 q^{49} - 16 q^{51} - 2 q^{53} + 2 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{69} + 16 q^{75} + 2 q^{77} + 22 q^{79} - 22 q^{81} + 20 q^{87} - 6 q^{91} - 10 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2157.1
1.00000i
1.00000i
0 2.00000 0 1.00000i 0 1.00000i 0 1.00000 0
2157.2 0 2.00000 0 1.00000i 0 1.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4004.2.m.a 2
13.b even 2 1 inner 4004.2.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.m.a 2 1.a even 1 1 trivial
4004.2.m.a 2 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(4004, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 13 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 25 \) Copy content Toggle raw display
$23$ \( (T + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 49 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T + 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1 \) Copy content Toggle raw display
$53$ \( (T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 9 \) Copy content Toggle raw display
$79$ \( (T - 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 81 \) Copy content Toggle raw display
$89$ \( T^{2} + 225 \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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