Properties

Label 1045.1.w.f
Level $1045$
Weight $1$
Character orbit 1045.w
Analytic conductor $0.522$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
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: \(16\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{40}^{7} + \zeta_{40}) q^{2} + (\zeta_{40}^{13} + \zeta_{40}^{11}) q^{3} + (\zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{4}+ \cdots + ( - \zeta_{40}^{6} + \cdots - \zeta_{40}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{40}^{7} + \zeta_{40}) q^{2} + (\zeta_{40}^{13} + \zeta_{40}^{11}) q^{3} + (\zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{4}+ \cdots + (\zeta_{40}^{18} + \zeta_{40}^{16} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} + 4 q^{5} - 12 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} + 4 q^{5} - 12 q^{6} - 4 q^{9} + 4 q^{16} + 4 q^{19} + 4 q^{20} - 20 q^{24} - 4 q^{25} - 8 q^{26} - 8 q^{30} + 16 q^{36} + 8 q^{39} - 16 q^{45} - 4 q^{49} + 40 q^{54} + 4 q^{64} - 12 q^{74} - 16 q^{76} + 16 q^{80} + 4 q^{81} - 4 q^{95} + 12 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(\zeta_{40}^{8}\) \(-1\)

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
−0.453990 + 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
0.453990 + 0.891007i
−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
0.987688 0.156434i
−0.610425 + 1.87869i −0.734572 + 0.533698i −2.34786 1.70582i −0.309017 0.951057i −0.554254 1.70582i 0 3.03979 2.20854i −0.0542543 + 0.166977i 1.97538
284.2 −0.0966818 + 0.297556i 1.44168 1.04744i 0.729825 + 0.530249i −0.309017 0.951057i 0.172288 + 0.530249i 0 −0.481456 + 0.349798i 0.672288 2.06909i 0.312869
284.3 0.0966818 0.297556i −1.44168 + 1.04744i 0.729825 + 0.530249i −0.309017 0.951057i 0.172288 + 0.530249i 0 0.481456 0.349798i 0.672288 2.06909i −0.312869
284.4 0.610425 1.87869i 0.734572 0.533698i −2.34786 1.70582i −0.309017 0.951057i −0.554254 1.70582i 0 −3.03979 + 2.20854i −0.0542543 + 0.166977i −1.97538
379.1 −0.610425 1.87869i −0.734572 0.533698i −2.34786 + 1.70582i −0.309017 + 0.951057i −0.554254 + 1.70582i 0 3.03979 + 2.20854i −0.0542543 0.166977i 1.97538
379.2 −0.0966818 0.297556i 1.44168 + 1.04744i 0.729825 0.530249i −0.309017 + 0.951057i 0.172288 0.530249i 0 −0.481456 0.349798i 0.672288 + 2.06909i 0.312869
379.3 0.0966818 + 0.297556i −1.44168 1.04744i 0.729825 0.530249i −0.309017 + 0.951057i 0.172288 0.530249i 0 0.481456 + 0.349798i 0.672288 + 2.06909i −0.312869
379.4 0.610425 + 1.87869i 0.734572 + 0.533698i −2.34786 + 1.70582i −0.309017 + 0.951057i −0.554254 + 1.70582i 0 −3.03979 2.20854i −0.0542543 0.166977i −1.97538
664.1 −1.44168 1.04744i 0.610425 1.87869i 0.672288 + 2.06909i 0.809017 0.587785i −2.84786 + 2.06909i 0 0.647354 1.99235i −2.34786 1.70582i −1.78201
664.2 −0.734572 0.533698i −0.0966818 + 0.297556i −0.0542543 0.166977i 0.809017 0.587785i 0.229825 0.166977i 0 −0.329843 + 1.01515i 0.729825 + 0.530249i −0.907981
664.3 0.734572 + 0.533698i 0.0966818 0.297556i −0.0542543 0.166977i 0.809017 0.587785i 0.229825 0.166977i 0 0.329843 1.01515i 0.729825 + 0.530249i 0.907981
664.4 1.44168 + 1.04744i −0.610425 + 1.87869i 0.672288 + 2.06909i 0.809017 0.587785i −2.84786 + 2.06909i 0 −0.647354 + 1.99235i −2.34786 1.70582i 1.78201
949.1 −1.44168 + 1.04744i 0.610425 + 1.87869i 0.672288 2.06909i 0.809017 + 0.587785i −2.84786 2.06909i 0 0.647354 + 1.99235i −2.34786 + 1.70582i −1.78201
949.2 −0.734572 + 0.533698i −0.0966818 0.297556i −0.0542543 + 0.166977i 0.809017 + 0.587785i 0.229825 + 0.166977i 0 −0.329843 1.01515i 0.729825 0.530249i −0.907981
949.3 0.734572 0.533698i 0.0966818 + 0.297556i −0.0542543 + 0.166977i 0.809017 + 0.587785i 0.229825 + 0.166977i 0 0.329843 + 1.01515i 0.729825 0.530249i 0.907981
949.4 1.44168 1.04744i −0.610425 1.87869i 0.672288 2.06909i 0.809017 + 0.587785i −2.84786 2.06909i 0 −0.647354 1.99235i −2.34786 + 1.70582i 1.78201
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 284.4
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.f 16
5.b even 2 1 inner 1045.1.w.f 16
11.c even 5 1 inner 1045.1.w.f 16
19.b odd 2 1 inner 1045.1.w.f 16
55.j even 10 1 inner 1045.1.w.f 16
95.d odd 2 1 CM 1045.1.w.f 16
209.m odd 10 1 inner 1045.1.w.f 16
1045.w odd 10 1 inner 1045.1.w.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.f 16 1.a even 1 1 trivial
1045.1.w.f 16 5.b even 2 1 inner
1045.1.w.f 16 11.c even 5 1 inner
1045.1.w.f 16 19.b odd 2 1 inner
1045.1.w.f 16 55.j even 10 1 inner
1045.1.w.f 16 95.d odd 2 1 CM
1045.1.w.f 16 209.m odd 10 1 inner
1045.1.w.f 16 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 \(S_{1}^{\mathrm{new}}(1045, [\chi])\):

\( T_{2}^{16} + 4T_{2}^{14} + 12T_{2}^{12} + 32T_{2}^{10} + 150T_{2}^{8} - 32T_{2}^{6} + 97T_{2}^{4} + 16T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) 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
$97$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
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