Properties

Label 2013.1.bm.c
Level $2013$
Weight $1$
Character orbit 2013.bm
Analytic conductor $1.005$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -183
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2013,1,Mod(548,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 6, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.548");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2013.bm (of order \(10\), degree \(4\), 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: \(1.00461787043\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.721218892933132803.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{2} + \zeta_{20}^{8} q^{3} + ( - \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{4} + ( - \zeta_{20}^{5} - \zeta_{20}^{3}) q^{6} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}) q^{8} - \zeta_{20}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{2} + \zeta_{20}^{8} q^{3} + ( - \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{4} + ( - \zeta_{20}^{5} - \zeta_{20}^{3}) q^{6} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}) q^{8} - \zeta_{20}^{6} q^{9} - \zeta_{20}^{7} q^{11} + ( - \zeta_{20}^{8} + \zeta_{20}^{2} + 1) q^{12} + (\zeta_{20}^{2} - 1) q^{13} + (\zeta_{20}^{8} + \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{16} + (\zeta_{20}^{7} + \zeta_{20}) q^{17} + (\zeta_{20}^{3} + \zeta_{20}) q^{18} + (\zeta_{20}^{6} - 1) q^{19} + (\zeta_{20}^{4} + \zeta_{20}^{2}) q^{22} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{23} + (\zeta_{20}^{9} + \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{3}) q^{24} + \zeta_{20}^{8} q^{25} + (\zeta_{20}^{9} - \zeta_{20}^{5}) q^{26} + \zeta_{20}^{4} q^{27} + ( - \zeta_{20}^{3} - \zeta_{20}) q^{29} + (\zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}^{3} + \zeta_{20}) q^{32} + \zeta_{20}^{5} q^{33} + (\zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2}) q^{34} + (\zeta_{20}^{8} + \zeta_{20}^{6} - 1) q^{36} + ( - \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{38} + ( - \zeta_{20}^{8} - 1) q^{39} + (\zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}) q^{44} + (\zeta_{20}^{8} + \zeta_{20}^{6} + \zeta_{20}^{4}) q^{46} + (\zeta_{20}^{8} - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{48} + \zeta_{20}^{4} q^{49} + ( - \zeta_{20}^{5} - \zeta_{20}^{3}) q^{50} + (\zeta_{20}^{9} - \zeta_{20}^{5}) q^{51} + ( - \zeta_{20}^{6} + 1) q^{52} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{53} + (\zeta_{20}^{9} - \zeta_{20}) q^{54} + ( - \zeta_{20}^{8} - \zeta_{20}^{4}) q^{57} + ( - 2 \zeta_{20}^{8} - \zeta_{20}^{6} + 1) q^{58} + ( - \zeta_{20}^{9} + \zeta_{20}^{5}) q^{59} + \zeta_{20}^{4} q^{61} + ( - \zeta_{20}^{8} - \zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{64} + ( - \zeta_{20}^{2} - 1) q^{66} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}) q^{68} + (\zeta_{20}^{9} + \zeta_{20}^{7}) q^{69} + ( - \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{72} + (\zeta_{20}^{8} - \zeta_{20}^{6}) q^{73} - \zeta_{20}^{6} q^{75} + ( - \zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{76} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{78} - \zeta_{20}^{2} q^{81} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{87} + ( - \zeta_{20}^{8} - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2}) q^{88} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{89} + (\zeta_{20}^{9} - \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}) q^{92} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}) q^{96} + \zeta_{20}^{6} q^{97} + (\zeta_{20}^{9} - \zeta_{20}) q^{98} - \zeta_{20}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 8 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} - 8 q^{4} - 2 q^{9} + 12 q^{12} - 6 q^{13} + 8 q^{16} - 6 q^{19} - 2 q^{25} - 2 q^{27} - 8 q^{36} - 6 q^{39} - 12 q^{48} - 2 q^{49} + 6 q^{52} + 4 q^{57} + 10 q^{58} - 2 q^{61} + 2 q^{64} - 10 q^{66} - 4 q^{73} - 2 q^{75} + 16 q^{76} - 2 q^{81} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2013\mathbb{Z}\right)^\times\).

\(n\) \(1222\) \(1343\) \(1465\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{20}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
548.1
0.951057 + 0.309017i
−0.951057 0.309017i
0.951057 0.309017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 + 1.80902i −0.809017 + 0.587785i −2.11803 1.53884i 0 −0.587785 1.80902i 0 2.48990 1.80902i 0.309017 0.951057i 0
548.2 0.587785 1.80902i −0.809017 + 0.587785i −2.11803 1.53884i 0 0.587785 + 1.80902i 0 −2.48990 + 1.80902i 0.309017 0.951057i 0
731.1 −0.587785 1.80902i −0.809017 0.587785i −2.11803 + 1.53884i 0 −0.587785 + 1.80902i 0 2.48990 + 1.80902i 0.309017 + 0.951057i 0
731.2 0.587785 + 1.80902i −0.809017 0.587785i −2.11803 + 1.53884i 0 0.587785 1.80902i 0 −2.48990 1.80902i 0.309017 + 0.951057i 0
1280.1 −0.951057 0.690983i 0.309017 0.951057i 0.118034 + 0.363271i 0 −0.951057 + 0.690983i 0 −0.224514 + 0.690983i −0.809017 0.587785i 0
1280.2 0.951057 + 0.690983i 0.309017 0.951057i 0.118034 + 0.363271i 0 0.951057 0.690983i 0 0.224514 0.690983i −0.809017 0.587785i 0
1829.1 −0.951057 + 0.690983i 0.309017 + 0.951057i 0.118034 0.363271i 0 −0.951057 0.690983i 0 −0.224514 0.690983i −0.809017 + 0.587785i 0
1829.2 0.951057 0.690983i 0.309017 + 0.951057i 0.118034 0.363271i 0 0.951057 + 0.690983i 0 0.224514 + 0.690983i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 548.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
183.d odd 2 1 CM by \(\Q(\sqrt{-183}) \)
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner
61.b even 2 1 inner
671.x even 10 1 inner
2013.bm odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2013.1.bm.c 8
3.b odd 2 1 inner 2013.1.bm.c 8
11.c even 5 1 inner 2013.1.bm.c 8
33.h odd 10 1 inner 2013.1.bm.c 8
61.b even 2 1 inner 2013.1.bm.c 8
183.d odd 2 1 CM 2013.1.bm.c 8
671.x even 10 1 inner 2013.1.bm.c 8
2013.bm odd 10 1 inner 2013.1.bm.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.1.bm.c 8 1.a even 1 1 trivial
2013.1.bm.c 8 3.b odd 2 1 inner
2013.1.bm.c 8 11.c even 5 1 inner
2013.1.bm.c 8 33.h odd 10 1 inner
2013.1.bm.c 8 61.b even 2 1 inner
2013.1.bm.c 8 183.d odd 2 1 CM
2013.1.bm.c 8 671.x even 10 1 inner
2013.1.bm.c 8 2013.bm odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5T_{2}^{6} + 10T_{2}^{4} + 25 \) acting on \(S_{1}^{\mathrm{new}}(2013, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) 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
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
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