Properties

Label 2601.1.p.a
Level $2601$
Weight $1$
Character orbit 2601.p
Analytic conductor $1.298$
Analytic rank $0$
Dimension $8$
Projective image $D_{2}$
CM/RM discs -3, -51, 17
Inner twists $32$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2601,1,Mod(802,2601)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2601.802");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2601.p (of order \(16\), degree \(8\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{17})\)
Artin image: $\OD_{64}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{6} q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{6} q^{4} - \zeta_{16}^{2} q^{13} - \zeta_{16}^{4} q^{16} + \zeta_{16}^{7} q^{19} + \zeta_{16}^{5} q^{25} - \zeta_{16} q^{43} - \zeta_{16}^{3} q^{49} + 2 q^{52} + \zeta_{16}^{2} q^{64} - \zeta_{16}^{4} q^{67} - 2 \zeta_{16}^{5} q^{76} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{52}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(290\) \(2026\)
\(\chi(n)\) \(1\) \(-\zeta_{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
802.1
−0.923880 0.382683i
−0.382683 0.923880i
0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
−0.923880 + 0.382683i
0.382683 + 0.923880i
0.923880 + 0.382683i
0 0 −0.707107 + 0.707107i 0 0 0 0 0 0
1081.1 0 0 0.707107 + 0.707107i 0 0 0 0 0 0
1405.1 0 0 −0.707107 0.707107i 0 0 0 0 0 0
1576.1 0 0 0.707107 0.707107i 0 0 0 0 0 0
1603.1 0 0 0.707107 0.707107i 0 0 0 0 0 0
1774.1 0 0 −0.707107 0.707107i 0 0 0 0 0 0
2098.1 0 0 0.707107 + 0.707107i 0 0 0 0 0 0
2377.1 0 0 −0.707107 + 0.707107i 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 802.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
51.c odd 2 1 CM by \(\Q(\sqrt{-51}) \)
17.c even 4 2 inner
17.d even 8 4 inner
17.e odd 16 8 inner
51.f odd 4 2 inner
51.g odd 8 4 inner
51.i even 16 8 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 256 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + 256 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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