Properties

Label 2008.1.j.a
Level $2008$
Weight $1$
Character orbit 2008.j
Analytic conductor $1.002$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2008,1,Mod(219,2008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2008, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2008.219");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2008 = 2^{3} \cdot 251 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2008.j (of order \(10\), degree \(4\), 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: \(1.00212254537\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.254024064064.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{2} + (\zeta_{10}^{2} - \zeta_{10}) q^{3} + q^{4} + (\zeta_{10}^{2} - \zeta_{10}) q^{6} + q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\zeta_{10}^{2} - \zeta_{10}) q^{3} + q^{4} + (\zeta_{10}^{2} - \zeta_{10}) q^{6} + q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{9} + ( - \zeta_{10}^{3} + 1) q^{11} + (\zeta_{10}^{2} - \zeta_{10}) q^{12} + q^{16} - \zeta_{10} q^{17} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{18} + \zeta_{10}^{2} q^{19} + ( - \zeta_{10}^{3} + 1) q^{22} + (\zeta_{10}^{2} - \zeta_{10}) q^{24} + q^{25} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} - 1) q^{27} + q^{32} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{33} - 2 \zeta_{10} q^{34} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{36} + 2 \zeta_{10}^{2} q^{38} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{41} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{43} + ( - \zeta_{10}^{3} + 1) q^{44} + (\zeta_{10}^{2} - \zeta_{10}) q^{48} + \zeta_{10}^{4} q^{49} + q^{50} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{51} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{54} + (2 \zeta_{10}^{4} - 2 \zeta_{10}^{3}) q^{57} + ( - \zeta_{10}^{3} + 1) q^{59} + q^{64} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{66} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{67} - 2 \zeta_{10} q^{68} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{72} - \zeta_{10} q^{73} + (\zeta_{10}^{2} - \zeta_{10}) q^{75} + 2 \zeta_{10}^{2} q^{76} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{81} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{82} + (\zeta_{10}^{2} + 1) q^{83} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{86} + ( - \zeta_{10}^{3} + 1) q^{88} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{89} + (\zeta_{10}^{2} - \zeta_{10}) q^{96} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{97} + \zeta_{10}^{4} q^{98} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} + 4 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} + 4 q^{8} - 3 q^{9} + 3 q^{11} - 2 q^{12} + 4 q^{16} - 2 q^{17} - 3 q^{18} - 2 q^{19} + 3 q^{22} - 2 q^{24} + 4 q^{25} + q^{27} + 4 q^{32} + q^{33} - 2 q^{34} - 3 q^{36} - 2 q^{38} - 2 q^{41} - 2 q^{43} + 3 q^{44} - 2 q^{48} - q^{49} + 4 q^{50} - 4 q^{51} + q^{54} - 4 q^{57} + 3 q^{59} + 4 q^{64} + q^{66} - 2 q^{67} - 2 q^{68} - 3 q^{72} - 2 q^{73} - 2 q^{75} - 2 q^{76} - 2 q^{82} + 3 q^{83} - 2 q^{86} + 3 q^{88} - 2 q^{89} - 2 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(503\) \(1005\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
219.1
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
1.00000 −0.500000 0.363271i 1.00000 0 −0.500000 0.363271i 0 1.00000 −0.190983 0.587785i 0
651.1 1.00000 −0.500000 + 0.363271i 1.00000 0 −0.500000 + 0.363271i 0 1.00000 −0.190983 + 0.587785i 0
1275.1 1.00000 −0.500000 1.53884i 1.00000 0 −0.500000 1.53884i 0 1.00000 −1.30902 + 0.951057i 0
1619.1 1.00000 −0.500000 + 1.53884i 1.00000 0 −0.500000 + 1.53884i 0 1.00000 −1.30902 0.951057i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
251.c even 5 1 inner
2008.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2008.1.j.a 4
8.d odd 2 1 CM 2008.1.j.a 4
251.c even 5 1 inner 2008.1.j.a 4
2008.j odd 10 1 inner 2008.1.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.1.j.a 4 1.a even 1 1 trivial
2008.1.j.a 4 8.d odd 2 1 CM
2008.1.j.a 4 251.c even 5 1 inner
2008.1.j.a 4 2008.j odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
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