Properties

Label 2001.1.i.b
Level $2001$
Weight $1$
Character orbit 2001.i
Analytic conductor $0.999$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2001,1,Mod(1172,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2001.1172");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2001.i (of order \(4\), degree \(2\), 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: \(0.998629090279\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.5048523.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} + i q^{3} - i q^{4} + (i + 1) q^{6} + q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} + i q^{3} - i q^{4} + (i + 1) q^{6} + q^{8} - q^{9} + q^{12} + i q^{13} + q^{16} + (i - 1) q^{18} + i q^{23} + q^{25} + (2 i + 2) q^{26} - i q^{27} - i q^{29} + ( - i - 1) q^{31} + ( - i + 1) q^{32} + i q^{36} - 2 q^{39} + ( - i - 1) q^{41} + (i + 1) q^{46} + (i + 1) q^{47} + i q^{48} + q^{49} + ( - i + 1) q^{50} + 2 q^{52} + ( - i - 1) q^{54} + ( - i - 1) q^{58} - i q^{59} - q^{62} - i q^{64} - q^{69} + (i - 1) q^{73} + i q^{75} + (2 i - 2) q^{78} + q^{81} - q^{82} + q^{87} + q^{92} + ( - i + 1) q^{93} + q^{94} + (i + 1) q^{96} + ( - i + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{6} - 2 q^{9} + 2 q^{12} + 2 q^{16} - 2 q^{18} + 2 q^{25} + 4 q^{26} - 2 q^{31} + 2 q^{32} - 4 q^{39} - 2 q^{41} + 2 q^{46} + 2 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{52} - 2 q^{54} - 2 q^{58} - 4 q^{62} - 2 q^{69} - 2 q^{73} - 4 q^{78} + 2 q^{81} - 4 q^{82} + 2 q^{87} + 2 q^{92} + 2 q^{93} + 4 q^{94} + 2 q^{96} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times\).

\(n\) \(553\) \(668\) \(1132\)
\(\chi(n)\) \(i\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1172.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000i 1.00000i 0 1.00000 + 1.00000i 0 0 −1.00000 0
1931.1 1.00000 + 1.00000i 1.00000i 1.00000i 0 1.00000 1.00000i 0 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
87.f even 4 1 inner
2001.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.1.i.b yes 2
3.b odd 2 1 2001.1.i.a 2
23.b odd 2 1 CM 2001.1.i.b yes 2
29.c odd 4 1 2001.1.i.a 2
69.c even 2 1 2001.1.i.a 2
87.f even 4 1 inner 2001.1.i.b yes 2
667.f even 4 1 2001.1.i.a 2
2001.i odd 4 1 inner 2001.1.i.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.1.i.a 2 3.b odd 2 1
2001.1.i.a 2 29.c odd 4 1
2001.1.i.a 2 69.c even 2 1
2001.1.i.a 2 667.f even 4 1
2001.1.i.b yes 2 1.a even 1 1 trivial
2001.1.i.b yes 2 23.b odd 2 1 CM
2001.1.i.b yes 2 87.f even 4 1 inner
2001.1.i.b yes 2 2001.i odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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