Properties

Label 2601.1.b.a
Level $2601$
Weight $1$
Character orbit 2601.b
Analytic conductor $1.298$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2601,1,Mod(2024,2601)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2601.2024");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2601.b (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: \(1.29806809786\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.7803.1
Artin image: $\GL(2,3)$
Artin field: Galois closure of 8.2.52788863403.1

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{4} - \beta q^{5} - q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{4} - \beta q^{5} - q^{7} - 2 q^{10} + q^{13} + \beta q^{14} - q^{16} - q^{19} + \beta q^{20} - \beta q^{23} - q^{25} - \beta q^{26} + q^{28} - q^{31} + \beta q^{32} + \beta q^{35} - q^{37} + \beta q^{38} + q^{43} - 2 q^{46} + \beta q^{50} - q^{52} + q^{61} + \beta q^{62} + q^{64} - \beta q^{65} + q^{67} + 2 q^{70} + \beta q^{71} + \beta q^{74} + q^{76} + \beta q^{80} - \beta q^{83} - \beta q^{86} + \beta q^{89} - q^{91} + \beta q^{92} + \beta q^{95} + q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{7} - 4 q^{10} + 2 q^{13} - 2 q^{16} - 2 q^{19} - 2 q^{25} + 2 q^{28} - 2 q^{31} - 2 q^{37} + 2 q^{43} - 4 q^{46} - 2 q^{52} + 2 q^{61} + 2 q^{64} + 2 q^{67} + 4 q^{70} + 2 q^{76} - 2 q^{91} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2601\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(2026\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2024.1
1.41421i
1.41421i
1.41421i 0 −1.00000 1.41421i 0 −1.00000 0 0 −2.00000
2024.2 1.41421i 0 −1.00000 1.41421i 0 −1.00000 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.1.b.a 2
3.b odd 2 1 inner 2601.1.b.a 2
17.b even 2 1 2601.1.b.b yes 2
17.c even 4 2 2601.1.c.a 4
17.d even 8 2 2601.1.g.a 4
17.d even 8 2 2601.1.g.b 4
17.e odd 16 4 2601.1.k.a 8
17.e odd 16 4 2601.1.k.b 8
51.c odd 2 1 2601.1.b.b yes 2
51.f odd 4 2 2601.1.c.a 4
51.g odd 8 2 2601.1.g.a 4
51.g odd 8 2 2601.1.g.b 4
51.i even 16 4 2601.1.k.a 8
51.i even 16 4 2601.1.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2601.1.b.a 2 1.a even 1 1 trivial
2601.1.b.a 2 3.b odd 2 1 inner
2601.1.b.b yes 2 17.b even 2 1
2601.1.b.b yes 2 51.c odd 2 1
2601.1.c.a 4 17.c even 4 2
2601.1.c.a 4 51.f odd 4 2
2601.1.g.a 4 17.d even 8 2
2601.1.g.a 4 51.g odd 8 2
2601.1.g.b 4 17.d even 8 2
2601.1.g.b 4 51.g odd 8 2
2601.1.k.a 8 17.e odd 16 4
2601.1.k.a 8 51.i even 16 4
2601.1.k.b 8 17.e odd 16 4
2601.1.k.b 8 51.i even 16 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2601, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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