Properties

Label 2023.1.d.b
Level $2023$
Weight $1$
Character orbit 2023.d
Analytic conductor $1.010$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2023,1,Mod(2022,2023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2023, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2023.2022");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2023.d (of order \(2\), degree \(1\), 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: \(1.00960852056\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.419904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 6x^{4} + 9x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.16748793615841.1

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{2}) q^{2} + ( - \beta_{2} + 1) q^{4} + \beta_{3} q^{7} + (\beta_{4} - \beta_{2} + 1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{2}) q^{2} + ( - \beta_{2} + 1) q^{4} + \beta_{3} q^{7} + (\beta_{4} - \beta_{2} + 1) q^{8} - q^{9} - \beta_{3} q^{11} - \beta_1 q^{14} + (\beta_{4} - \beta_{2} + 1) q^{16} + ( - \beta_{4} + \beta_{2}) q^{18} + \beta_1 q^{22} + \beta_{5} q^{23} - q^{25} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{28} + (\beta_{5} + \beta_1) q^{29} + ( - \beta_{2} + 1) q^{32} + (\beta_{2} - 1) q^{36} - \beta_1 q^{37} - \beta_{4} q^{43} + (\beta_{5} - \beta_{3} + \beta_1) q^{44} + ( - \beta_{5} + \beta_{3}) q^{46} - q^{49} + ( - \beta_{4} + \beta_{2}) q^{50} - \beta_{4} q^{53} + (\beta_{3} - \beta_1) q^{56} + ( - \beta_{3} + \beta_1) q^{58} - \beta_{3} q^{63} + (\beta_{4} - \beta_{2}) q^{64} - q^{67} + (\beta_{5} + \beta_1) q^{71} + ( - \beta_{4} + \beta_{2} - 1) q^{72} + ( - \beta_{5} + 2 \beta_{3} - \beta_1) q^{74} + q^{77} + \beta_1 q^{79} + q^{81} + (\beta_{4} - 1) q^{86} + ( - \beta_{3} + \beta_1) q^{88} + ( - \beta_{3} - \beta_1) q^{92} + ( - \beta_{4} + \beta_{2}) q^{98} + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{4} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{4} + 6 q^{8} - 6 q^{9} + 6 q^{16} - 6 q^{25} + 6 q^{32} - 6 q^{36} - 6 q^{49} - 6 q^{67} - 6 q^{72} + 6 q^{77} + 6 q^{81} - 6 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 6x^{4} + 9x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(1737\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2022.1
1.53209i
1.53209i
0.347296i
0.347296i
1.87939i
1.87939i
−1.53209 0 1.34730 0 0 1.00000i −0.532089 −1.00000 0
2022.2 −1.53209 0 1.34730 0 0 1.00000i −0.532089 −1.00000 0
2022.3 −0.347296 0 −0.879385 0 0 1.00000i 0.652704 −1.00000 0
2022.4 −0.347296 0 −0.879385 0 0 1.00000i 0.652704 −1.00000 0
2022.5 1.87939 0 2.53209 0 0 1.00000i 2.87939 −1.00000 0
2022.6 1.87939 0 2.53209 0 0 1.00000i 2.87939 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2022.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
17.b even 2 1 inner
119.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2023.1.d.b 6
7.b odd 2 1 CM 2023.1.d.b 6
17.b even 2 1 inner 2023.1.d.b 6
17.c even 4 1 2023.1.c.c 3
17.c even 4 1 2023.1.c.d yes 3
17.d even 8 4 2023.1.f.c 12
17.e odd 16 8 2023.1.l.c 24
119.d odd 2 1 inner 2023.1.d.b 6
119.f odd 4 1 2023.1.c.c 3
119.f odd 4 1 2023.1.c.d yes 3
119.l odd 8 4 2023.1.f.c 12
119.p even 16 8 2023.1.l.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2023.1.c.c 3 17.c even 4 1
2023.1.c.c 3 119.f odd 4 1
2023.1.c.d yes 3 17.c even 4 1
2023.1.c.d yes 3 119.f odd 4 1
2023.1.d.b 6 1.a even 1 1 trivial
2023.1.d.b 6 7.b odd 2 1 CM
2023.1.d.b 6 17.b even 2 1 inner
2023.1.d.b 6 119.d odd 2 1 inner
2023.1.f.c 12 17.d even 8 4
2023.1.f.c 12 119.l odd 8 4
2023.1.l.c 24 17.e odd 16 8
2023.1.l.c 24 119.p even 16 8

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(2023, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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