Properties

Label 2997.1.n.d
Level $2997$
Weight $1$
Character orbit 2997.n
Analytic conductor $1.496$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -111
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2997,1,Mod(998,2997)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2997, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2997.998");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2997 = 3^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2997.n (of order \(6\), degree \(2\), not 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: \(1.49569784286\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.4107.1
Artin image: $C_3\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} + \cdots)\)

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{5} - 2 q^{10} + (\beta_{2} + 1) q^{16} - \beta_{3} q^{17} + \beta_1 q^{20} + ( - \beta_{3} - \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} - \beta_1 q^{29} + ( - \beta_{3} - \beta_1) q^{32} + ( - 2 \beta_{2} - 2) q^{34} - q^{37} - 2 q^{46} - \beta_{2} q^{49} + (\beta_{3} + \beta_1) q^{50} + 2 \beta_{2} q^{58} + (\beta_{3} + \beta_1) q^{59} - q^{64} + (\beta_{3} + \beta_1) q^{68} + \beta_1 q^{74} - \beta_{3} q^{80} - 2 \beta_{2} q^{85} + \beta_{3} q^{89} + \beta_1 q^{92} + \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} - 8 q^{10} + 2 q^{16} - 2 q^{25} - 4 q^{34} - 4 q^{37} - 8 q^{46} + 2 q^{49} - 4 q^{58} - 4 q^{64} + 4 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2997\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1703\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
998.1
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
−0.707107 + 1.22474i 0 −0.500000 0.866025i 0.707107 + 1.22474i 0 0 0 0 −2.00000
998.2 0.707107 1.22474i 0 −0.500000 0.866025i −0.707107 1.22474i 0 0 0 0 −2.00000
1997.1 −0.707107 1.22474i 0 −0.500000 + 0.866025i 0.707107 1.22474i 0 0 0 0 −2.00000
1997.2 0.707107 + 1.22474i 0 −0.500000 + 0.866025i −0.707107 + 1.22474i 0 0 0 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
111.d odd 2 1 CM by \(\Q(\sqrt{-111}) \)
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
37.b even 2 1 inner
333.n odd 6 1 inner
333.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2997.1.n.d 4
3.b odd 2 1 inner 2997.1.n.d 4
9.c even 3 1 111.1.d.b 2
9.c even 3 1 inner 2997.1.n.d 4
9.d odd 6 1 111.1.d.b 2
9.d odd 6 1 inner 2997.1.n.d 4
36.f odd 6 1 1776.1.n.c 2
36.h even 6 1 1776.1.n.c 2
37.b even 2 1 inner 2997.1.n.d 4
45.h odd 6 1 2775.1.h.b 2
45.j even 6 1 2775.1.h.b 2
45.k odd 12 2 2775.1.b.b 4
45.l even 12 2 2775.1.b.b 4
111.d odd 2 1 CM 2997.1.n.d 4
333.n odd 6 1 111.1.d.b 2
333.n odd 6 1 inner 2997.1.n.d 4
333.q even 6 1 111.1.d.b 2
333.q even 6 1 inner 2997.1.n.d 4
1332.z even 6 1 1776.1.n.c 2
1332.bl odd 6 1 1776.1.n.c 2
1665.bh even 6 1 2775.1.h.b 2
1665.bt odd 6 1 2775.1.h.b 2
1665.dc odd 12 2 2775.1.b.b 4
1665.di even 12 2 2775.1.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.b 2 9.c even 3 1
111.1.d.b 2 9.d odd 6 1
111.1.d.b 2 333.n odd 6 1
111.1.d.b 2 333.q even 6 1
1776.1.n.c 2 36.f odd 6 1
1776.1.n.c 2 36.h even 6 1
1776.1.n.c 2 1332.z even 6 1
1776.1.n.c 2 1332.bl odd 6 1
2775.1.b.b 4 45.k odd 12 2
2775.1.b.b 4 45.l even 12 2
2775.1.b.b 4 1665.dc odd 12 2
2775.1.b.b 4 1665.di even 12 2
2775.1.h.b 2 45.h odd 6 1
2775.1.h.b 2 45.j even 6 1
2775.1.h.b 2 1665.bh even 6 1
2775.1.h.b 2 1665.bt odd 6 1
2997.1.n.d 4 1.a even 1 1 trivial
2997.1.n.d 4 3.b odd 2 1 inner
2997.1.n.d 4 9.c even 3 1 inner
2997.1.n.d 4 9.d odd 6 1 inner
2997.1.n.d 4 37.b even 2 1 inner
2997.1.n.d 4 111.d odd 2 1 CM
2997.1.n.d 4 333.n odd 6 1 inner
2997.1.n.d 4 333.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2997, [\chi])\):

\( T_{2}^{4} + 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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