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Properties

Label 1045.1.w.e
Level $1045$
Weight $1$
Character orbit 1045.w
Analytic conductor $0.522$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -95
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,1,Mod(284,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, 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("1045.284");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1045.w (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: \(0.521522938201\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.87298324158753125.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{9} - \zeta_{20}^{3}) q^{2} + (\zeta_{20}^{9} + \zeta_{20}^{7}) q^{3} + ( - \zeta_{20}^{8} + \cdots - \zeta_{20}^{2}) q^{4}+ \cdots + ( - \zeta_{20}^{8} + \cdots - \zeta_{20}^{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{9} - \zeta_{20}^{3}) q^{2} + (\zeta_{20}^{9} + \zeta_{20}^{7}) q^{3} + ( - \zeta_{20}^{8} + \cdots - \zeta_{20}^{2}) q^{4}+ \cdots + (\zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 2 q^{5} + 10 q^{6} + 2 q^{9} + 2 q^{11} - 12 q^{16} - 2 q^{19} - 8 q^{20} + 10 q^{24} - 2 q^{25} - 10 q^{30} - 2 q^{36} - 10 q^{39} - 2 q^{44} + 12 q^{45} - 2 q^{49} + 20 q^{54} - 8 q^{55} - 6 q^{61} - 8 q^{64} + 10 q^{74} + 12 q^{76} + 8 q^{80} - 12 q^{81} - 2 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
284.1
−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
−0.363271 + 1.11803i 1.53884 − 1.11803i −0.309017 − 0.224514i 0.309017 + 0.951057i 0.690983 + 2.12663i 0 −0.587785 + 0.427051i 0.809017 − 2.48990i −1.17557
284.2 0.363271 − 1.11803i −1.53884 + 1.11803i −0.309017 − 0.224514i 0.309017 + 0.951057i 0.690983 + 2.12663i 0 0.587785 − 0.427051i 0.809017 − 2.48990i 1.17557
379.1 −0.363271 − 1.11803i 1.53884 + 1.11803i −0.309017 + 0.224514i 0.309017 − 0.951057i 0.690983 − 2.12663i 0 −0.587785 − 0.427051i 0.809017 + 2.48990i −1.17557
379.2 0.363271 + 1.11803i −1.53884 − 1.11803i −0.309017 + 0.224514i 0.309017 − 0.951057i 0.690983 − 2.12663i 0 0.587785 + 0.427051i 0.809017 + 2.48990i 1.17557
664.1 −1.53884 − 1.11803i −0.363271 + 1.11803i 0.809017 + 2.48990i −0.809017 + 0.587785i 1.80902 − 1.31433i 0 0.951057 − 2.92705i −0.309017 − 0.224514i 1.90211
664.2 1.53884 + 1.11803i 0.363271 − 1.11803i 0.809017 + 2.48990i −0.809017 + 0.587785i 1.80902 − 1.31433i 0 −0.951057 + 2.92705i −0.309017 − 0.224514i −1.90211
949.1 −1.53884 + 1.11803i −0.363271 − 1.11803i 0.809017 − 2.48990i −0.809017 − 0.587785i 1.80902 + 1.31433i 0 0.951057 + 2.92705i −0.309017 + 0.224514i 1.90211
949.2 1.53884 − 1.11803i 0.363271 + 1.11803i 0.809017 − 2.48990i −0.809017 − 0.587785i 1.80902 + 1.31433i 0 −0.951057 − 2.92705i −0.309017 + 0.224514i −1.90211
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 284.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
11.c even 5 1 inner
19.b odd 2 1 inner
55.j even 10 1 inner
209.m odd 10 1 inner
1045.w odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.w.e ✓ 8
5.b even 2 1 inner 1045.1.w.e ✓ 8
11.c even 5 1 inner 1045.1.w.e ✓ 8
19.b odd 2 1 inner 1045.1.w.e ✓ 8
55.j even 10 1 inner 1045.1.w.e ✓ 8
95.d odd 2 1 CM 1045.1.w.e ✓ 8
209.m odd 10 1 inner 1045.1.w.e ✓ 8
1045.w odd 10 1 inner 1045.1.w.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.e ✓ 8 1.a even 1 1 trivial
1045.1.w.e ✓ 8 5.b even 2 1 inner
1045.1.w.e ✓ 8 11.c even 5 1 inner
1045.1.w.e ✓ 8 19.b odd 2 1 inner
1045.1.w.e ✓ 8 55.j even 10 1 inner
1045.1.w.e ✓ 8 95.d odd 2 1 CM
1045.1.w.e ✓ 8 209.m odd 10 1 inner
1045.1.w.e ✓ 8 1045.w odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1045, [\chi])\):

\( T_{2}^{8} + 10T_{2}^{4} + 25T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) 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
$37$ \( T^{8} + 10 T^{4} + \cdots + 25 \) 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
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{2} + 5)^{2} \) 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
$97$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
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