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Properties

Label 2541.1.r.e
Level $2541$
Weight $1$
Character orbit 2541.r
Analytic conductor $1.268$
Analytic rank $0$
Dimension $16$
Projective image $D_{4}$
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] = [2541,1,Mod(524,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.524");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2541.r (of order \(10\), degree \(4\), 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: \(1.26812419710\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.195657.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{40}^{14} q^{3} - \zeta_{40}^{16} q^{4} - \zeta_{40}^{11} q^{7} - \zeta_{40}^{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{40}^{14} q^{3} - \zeta_{40}^{16} q^{4} - \zeta_{40}^{11} q^{7} - \zeta_{40}^{8} q^{9} + \zeta_{40}^{10} q^{12} + (\zeta_{40}^{13} + \zeta_{40}^{3}) q^{13} - \zeta_{40}^{12} q^{16} + (\zeta_{40}^{19} - \zeta_{40}^{9}) q^{19} + \zeta_{40}^{5} q^{21} + \zeta_{40}^{4} q^{25} + \zeta_{40}^{2} q^{27} - \zeta_{40}^{7} q^{28} - \zeta_{40}^{18} q^{31} - \zeta_{40}^{4} q^{36} + (\zeta_{40}^{17} - \zeta_{40}^{7}) q^{39} + ( - \zeta_{40}^{15} - \zeta_{40}^{5}) q^{43} + \zeta_{40}^{6} q^{48} - \zeta_{40}^{2} q^{49} + ( - \zeta_{40}^{19} + \zeta_{40}^{9}) q^{52} + ( - \zeta_{40}^{13} + \zeta_{40}^{3}) q^{57} + ( - \zeta_{40}^{17} - \zeta_{40}^{7}) q^{61} + \zeta_{40}^{19} q^{63} - \zeta_{40}^{8} q^{64} + (\zeta_{40}^{11} - \zeta_{40}) q^{73} + \zeta_{40}^{18} q^{75} + (\zeta_{40}^{15} - \zeta_{40}^{5}) q^{76} + ( - \zeta_{40}^{13} + \zeta_{40}^{3}) q^{79} + \zeta_{40}^{16} q^{81} + \zeta_{40} q^{84} + ( - \zeta_{40}^{14} + \zeta_{40}^{4}) q^{91} + 2 \zeta_{40}^{12} q^{93} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 4 q^{9} - 4 q^{16} + 4 q^{25} - 4 q^{36} + 4 q^{64} - 4 q^{81} + 4 q^{91} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2541\mathbb{Z}\right)^\times\).

\(n\) \(848\) \(1816\) \(2059\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{40}^{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
0.453990 − 0.891007i
−0.453990 + 0.891007i
0.891007 + 0.453990i
−0.891007 − 0.453990i
0.453990 + 0.891007i
−0.453990 − 0.891007i
0.891007 − 0.453990i
−0.891007 + 0.453990i
−0.987688 + 0.156434i
0.987688 − 0.156434i
−0.156434 − 0.987688i
0.156434 + 0.987688i
−0.987688 − 0.156434i
0.987688 + 0.156434i
−0.156434 + 0.987688i
0.156434 − 0.987688i
0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 0 −0.891007 − 0.453990i 0 0.809017 + 0.587785i 0
524.2 0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 0 0.891007 + 0.453990i 0 0.809017 + 0.587785i 0
524.3 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 0 −0.453990 + 0.891007i 0 0.809017 + 0.587785i 0
524.4 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 0 0.453990 − 0.891007i 0 0.809017 + 0.587785i 0
965.1 0 −0.951057 + 0.309017i −0.309017 + 0.951057i 0 0 −0.891007 + 0.453990i 0 0.809017 − 0.587785i 0
965.2 0 −0.951057 + 0.309017i −0.309017 + 0.951057i 0 0 0.891007 − 0.453990i 0 0.809017 − 0.587785i 0
965.3 0 0.951057 − 0.309017i −0.309017 + 0.951057i 0 0 −0.453990 − 0.891007i 0 0.809017 − 0.587785i 0
965.4 0 0.951057 − 0.309017i −0.309017 + 0.951057i 0 0 0.453990 + 0.891007i 0 0.809017 − 0.587785i 0
1322.1 0 −0.587785 − 0.809017i 0.809017 + 0.587785i 0 0 −0.156434 − 0.987688i 0 −0.309017 + 0.951057i 0
1322.2 0 −0.587785 − 0.809017i 0.809017 + 0.587785i 0 0 0.156434 + 0.987688i 0 −0.309017 + 0.951057i 0
1322.3 0 0.587785 + 0.809017i 0.809017 + 0.587785i 0 0 −0.987688 + 0.156434i 0 −0.309017 + 0.951057i 0
1322.4 0 0.587785 + 0.809017i 0.809017 + 0.587785i 0 0 0.987688 − 0.156434i 0 −0.309017 + 0.951057i 0
2393.1 0 −0.587785 + 0.809017i 0.809017 − 0.587785i 0 0 −0.156434 + 0.987688i 0 −0.309017 − 0.951057i 0
2393.2 0 −0.587785 + 0.809017i 0.809017 − 0.587785i 0 0 0.156434 − 0.987688i 0 −0.309017 − 0.951057i 0
2393.3 0 0.587785 − 0.809017i 0.809017 − 0.587785i 0 0 −0.987688 − 0.156434i 0 −0.309017 − 0.951057i 0
2393.4 0 0.587785 − 0.809017i 0.809017 − 0.587785i 0 0 0.987688 + 0.156434i 0 −0.309017 − 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 524.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
21.c even 2 1 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner
77.b even 2 1 inner
77.j odd 10 3 inner
77.l even 10 3 inner
231.h odd 2 1 inner
231.r odd 10 3 inner
231.u even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2541.1.r.e 16
3.b odd 2 1 CM 2541.1.r.e 16
7.b odd 2 1 inner 2541.1.r.e 16
11.b odd 2 1 inner 2541.1.r.e 16
11.c even 5 1 2541.1.h.a ✓ 4
11.c even 5 3 inner 2541.1.r.e 16
11.d odd 10 1 2541.1.h.a ✓ 4
11.d odd 10 3 inner 2541.1.r.e 16
21.c even 2 1 inner 2541.1.r.e 16
33.d even 2 1 inner 2541.1.r.e 16
33.f even 10 1 2541.1.h.a ✓ 4
33.f even 10 3 inner 2541.1.r.e 16
33.h odd 10 1 2541.1.h.a ✓ 4
33.h odd 10 3 inner 2541.1.r.e 16
77.b even 2 1 inner 2541.1.r.e 16
77.j odd 10 1 2541.1.h.a ✓ 4
77.j odd 10 3 inner 2541.1.r.e 16
77.l even 10 1 2541.1.h.a ✓ 4
77.l even 10 3 inner 2541.1.r.e 16
231.h odd 2 1 inner 2541.1.r.e 16
231.r odd 10 1 2541.1.h.a ✓ 4
231.r odd 10 3 inner 2541.1.r.e 16
231.u even 10 1 2541.1.h.a ✓ 4
231.u even 10 3 inner 2541.1.r.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.1.h.a ✓ 4 11.c even 5 1
2541.1.h.a ✓ 4 11.d odd 10 1
2541.1.h.a ✓ 4 33.f even 10 1
2541.1.h.a ✓ 4 33.h odd 10 1
2541.1.h.a ✓ 4 77.j odd 10 1
2541.1.h.a ✓ 4 77.l even 10 1
2541.1.h.a ✓ 4 231.r odd 10 1
2541.1.h.a ✓ 4 231.u even 10 1
2541.1.r.e 16 1.a even 1 1 trivial
2541.1.r.e 16 3.b odd 2 1 CM
2541.1.r.e 16 7.b odd 2 1 inner
2541.1.r.e 16 11.b odd 2 1 inner
2541.1.r.e 16 11.c even 5 3 inner
2541.1.r.e 16 11.d odd 10 3 inner
2541.1.r.e 16 21.c even 2 1 inner
2541.1.r.e 16 33.d even 2 1 inner
2541.1.r.e 16 33.f even 10 3 inner
2541.1.r.e 16 33.h odd 10 3 inner
2541.1.r.e 16 77.b even 2 1 inner
2541.1.r.e 16 77.j odd 10 3 inner
2541.1.r.e 16 77.l even 10 3 inner
2541.1.r.e 16 231.h odd 2 1 inner
2541.1.r.e 16 231.r odd 10 3 inner
2541.1.r.e 16 231.u even 10 3 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2541, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2)^{8} \) 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
$61$ \( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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