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Properties

Label 2601.1.p.b
Level $2601$
Weight $1$
Character orbit 2601.p
Analytic conductor $1.298$
Analytic rank $0$
Dimension $16$
Projective image $D_{6}$
CM discriminant -3
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: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: 16.0.121029087867608368152576.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.12778713.1

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{4} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{4} + \beta_{3} q^{7} + \beta_{4} q^{13} - \beta_{8} q^{16} - \beta_{14} q^{19} + \beta_{10} q^{25} + \beta_{15} q^{28} - \beta_1 q^{31} - \beta_{9} q^{37} + \beta_{2} q^{43} + 2 \beta_{6} q^{49} - q^{52} + \beta_{11} q^{61} + \beta_{4} q^{64} + \beta_{8} q^{67} + \beta_{10} q^{76} + \beta_{7} q^{91} - \beta_{5} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{52}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 243 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 243 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{12} ) / 729 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{13} ) / 729 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{14} ) / 2187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} ) / 2187 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 81\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 243\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 243\beta_{11} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 729\beta_{12} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 729\beta_{13} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2187\beta_{14} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2187\beta_{15} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(290\) \(2026\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
802.1
0.337906 − 1.69877i
−0.337906 + 1.69877i
0.962276 − 1.44015i
−0.962276 + 1.44015i
−1.69877 + 0.337906i
1.69877 − 0.337906i
0.962276 + 1.44015i
−0.962276 − 1.44015i
1.44015 − 0.962276i
−1.44015 + 0.962276i
0.337906 + 1.69877i
−0.337906 − 1.69877i
1.44015 + 0.962276i
−1.44015 − 0.962276i
−1.69877 − 0.337906i
1.69877 + 0.337906i
0 0 −0.707107 + 0.707107i 0 0 −0.962276 + 1.44015i 0 0 0
802.2 0 0 −0.707107 + 0.707107i 0 0 0.962276 − 1.44015i 0 0 0
1081.1 0 0 0.707107 + 0.707107i 0 0 −1.69877 − 0.337906i 0 0 0
1081.2 0 0 0.707107 + 0.707107i 0 0 1.69877 + 0.337906i 0 0 0
1405.1 0 0 −0.707107 − 0.707107i 0 0 −1.44015 + 0.962276i 0 0 0
1405.2 0 0 −0.707107 − 0.707107i 0 0 1.44015 − 0.962276i 0 0 0
1576.1 0 0 0.707107 − 0.707107i 0 0 −1.69877 + 0.337906i 0 0 0
1576.2 0 0 0.707107 − 0.707107i 0 0 1.69877 − 0.337906i 0 0 0
1603.1 0 0 0.707107 − 0.707107i 0 0 −0.337906 − 1.69877i 0 0 0
1603.2 0 0 0.707107 − 0.707107i 0 0 0.337906 + 1.69877i 0 0 0
1774.1 0 0 −0.707107 − 0.707107i 0 0 −0.962276 − 1.44015i 0 0 0
1774.2 0 0 −0.707107 − 0.707107i 0 0 0.962276 + 1.44015i 0 0 0
2098.1 0 0 0.707107 + 0.707107i 0 0 −0.337906 + 1.69877i 0 0 0
2098.2 0 0 0.707107 + 0.707107i 0 0 0.337906 − 1.69877i 0 0 0
2377.1 0 0 −0.707107 + 0.707107i 0 0 −1.44015 − 0.962276i 0 0 0
2377.2 0 0 −0.707107 + 0.707107i 0 0 1.44015 + 0.962276i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 802.2
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 inner
17.c even 4 2 inner
17.d even 8 4 inner
17.e odd 16 8 inner
51.c odd 2 1 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.b ✓ 16
3.b odd 2 1 CM 2601.1.p.b ✓ 16
17.b even 2 1 inner 2601.1.p.b ✓ 16
17.c even 4 2 inner 2601.1.p.b ✓ 16
17.d even 8 4 inner 2601.1.p.b ✓ 16
17.e odd 16 8 inner 2601.1.p.b ✓ 16
51.c odd 2 1 inner 2601.1.p.b ✓ 16
51.f odd 4 2 inner 2601.1.p.b ✓ 16
51.g odd 8 4 inner 2601.1.p.b ✓ 16
51.i even 16 8 inner 2601.1.p.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2601.1.p.b ✓ 16 1.a even 1 1 trivial
2601.1.p.b ✓ 16 3.b odd 2 1 CM
2601.1.p.b ✓ 16 17.b even 2 1 inner
2601.1.p.b ✓ 16 17.c even 4 2 inner
2601.1.p.b ✓ 16 17.d even 8 4 inner
2601.1.p.b ✓ 16 17.e odd 16 8 inner
2601.1.p.b ✓ 16 51.c odd 2 1 inner
2601.1.p.b ✓ 16 51.f odd 4 2 inner
2601.1.p.b ✓ 16 51.g odd 8 4 inner
2601.1.p.b ✓ 16 51.i even 16 8 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 6561 \) acting on \(S_{1}^{\mathrm{new}}(2601, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 6561 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + 6561 \) Copy content Toggle raw display
$37$ \( T^{16} + 6561 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} + 6561 \) Copy content Toggle raw display
$67$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + 6561 \) Copy content Toggle raw display
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