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Properties

Label 1053.2.e.q
Level $1053$
Weight $2$
Character orbit 1053.e
Analytic conductor $8.408$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1053,2,Mod(352,1053)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1053, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1053.352");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1053 = 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1053.e (of order \(3\), degree \(2\), 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: \(8.40824733284\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.38862602496.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 46x^{4} - 21x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 351)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{4} + \beta_{2} - 1) q^{4} + ( - \beta_{5} + \beta_{3}) q^{5} - 2 \beta_{2} q^{7} + ( - 2 \beta_{6} + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{4} + \beta_{2} - 1) q^{4} + ( - \beta_{5} + \beta_{3}) q^{5} - 2 \beta_{2} q^{7} + ( - 2 \beta_{6} + \beta_{3}) q^{8} + \beta_1 q^{10} + 2 \beta_{7} q^{11} + (\beta_{2} - 1) q^{13} + (2 \beta_{7} + 2 \beta_{6}) q^{14} + (\beta_{4} - 4 \beta_{2} + \beta_1) q^{16} - 2 \beta_{6} q^{17} + (2 \beta_1 + 2) q^{19} + (3 \beta_{7} - \beta_{5}) q^{20} + (2 \beta_{4} - 6 \beta_{2} + 6) q^{22} + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots - 2 \beta_{3}) q^{23}+ \cdots + 3 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} - 8 q^{7} - 4 q^{10} - 4 q^{13} - 18 q^{16} + 8 q^{19} + 28 q^{22} - 12 q^{25} + 24 q^{28} - 12 q^{31} - 28 q^{34} + 24 q^{37} + 36 q^{40} + 8 q^{43} - 48 q^{46} + 12 q^{49} - 6 q^{52} + 8 q^{55} + 32 q^{58} + 8 q^{64} + 12 q^{67} + 4 q^{70} + 64 q^{73} + 68 q^{76} + 16 q^{79} - 48 q^{82} + 4 q^{85} + 60 q^{88} + 16 q^{91} - 6 q^{94} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 46x^{4} - 21x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 163 ) / 46 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 46\nu^{4} + 322\nu^{2} - 9 ) / 138 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{7} - 92\nu^{5} + 575\nu^{3} - 507\nu ) / 207 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{6} + 184\nu^{4} - 1150\nu^{2} + 525 ) / 138 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -31\nu^{7} + 184\nu^{5} - 1150\nu^{3} - 453\nu ) / 414 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -43\nu^{7} + 322\nu^{5} - 2116\nu^{3} + 1869\nu ) / 414 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -26\nu^{7} + 161\nu^{5} - 1058\nu^{3} - 420\nu ) / 207 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 4\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} - 4\beta_{6} + 14\beta_{5} - 7\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{4} + 25\beta_{2} - 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{7} - 50\beta_{6} + 46\beta_{5} - 92\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -46\beta _1 - 163 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 163\beta_{7} - 163\beta_{6} - 301\beta_{5} - 301\beta_{3} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(730\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
352.1
−0.586484 − 0.338607i
−2.21496 − 1.27881i
2.21496 + 1.27881i
0.586484 + 0.338607i
−0.586484 + 0.338607i
−2.21496 + 1.27881i
2.21496 − 1.27881i
0.586484 − 0.338607i
−1.27881 − 2.21496i 0 −2.27069 + 3.93295i 0.692322 − 1.19914i 0 −1.00000 − 1.73205i 6.49987 0 −3.54138
352.2 −0.338607 − 0.586484i 0 0.770691 − 1.33488i −1.87635 + 3.24993i 0 −1.00000 − 1.73205i −2.39827 0 2.54138
352.3 0.338607 + 0.586484i 0 0.770691 − 1.33488i 1.87635 − 3.24993i 0 −1.00000 − 1.73205i 2.39827 0 2.54138
352.4 1.27881 + 2.21496i 0 −2.27069 + 3.93295i −0.692322 + 1.19914i 0 −1.00000 − 1.73205i −6.49987 0 −3.54138
703.1 −1.27881 + 2.21496i 0 −2.27069 − 3.93295i 0.692322 + 1.19914i 0 −1.00000 + 1.73205i 6.49987 0 −3.54138
703.2 −0.338607 + 0.586484i 0 0.770691 + 1.33488i −1.87635 − 3.24993i 0 −1.00000 + 1.73205i −2.39827 0 2.54138
703.3 0.338607 − 0.586484i 0 0.770691 + 1.33488i 1.87635 + 3.24993i 0 −1.00000 + 1.73205i 2.39827 0 2.54138
703.4 1.27881 − 2.21496i 0 −2.27069 − 3.93295i −0.692322 − 1.19914i 0 −1.00000 + 1.73205i −6.49987 0 −3.54138
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 352.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1053.2.e.q 8
3.b odd 2 1 inner 1053.2.e.q 8
9.c even 3 1 351.2.a.e ✓ 4
9.c even 3 1 inner 1053.2.e.q 8
9.d odd 6 1 351.2.a.e ✓ 4
9.d odd 6 1 inner 1053.2.e.q 8
36.f odd 6 1 5616.2.a.cg 4
36.h even 6 1 5616.2.a.cg 4
45.h odd 6 1 8775.2.a.bk 4
45.j even 6 1 8775.2.a.bk 4
117.n odd 6 1 4563.2.a.z 4
117.t even 6 1 4563.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
351.2.a.e ✓ 4 9.c even 3 1
351.2.a.e ✓ 4 9.d odd 6 1
1053.2.e.q 8 1.a even 1 1 trivial
1053.2.e.q 8 3.b odd 2 1 inner
1053.2.e.q 8 9.c even 3 1 inner
1053.2.e.q 8 9.d odd 6 1 inner
4563.2.a.z 4 117.n odd 6 1
4563.2.a.z 4 117.t even 6 1
5616.2.a.cg 4 36.f odd 6 1
5616.2.a.cg 4 36.h even 6 1
8775.2.a.bk 4 45.h odd 6 1
8775.2.a.bk 4 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):

\( T_{2}^{8} + 7T_{2}^{6} + 46T_{2}^{4} + 21T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{8} + 16T_{5}^{6} + 229T_{5}^{4} + 432T_{5}^{2} + 729 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 7 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 28 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 28 T^{2} + 48)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 36)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 84 T^{6} + \cdots + 186624 \) Copy content Toggle raw display
$29$ \( T^{8} + 100 T^{6} + \cdots + 5531904 \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 28)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + 84 T^{6} + \cdots + 186624 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 144 T^{6} + \cdots + 4782969 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 120 T^{6} + \cdots + 10673289 \) Copy content Toggle raw display
$61$ \( (T^{4} + 37 T^{2} + 1369)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 48 T^{2} + 243)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 280 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( (T^{4} - 160 T^{2} + 147)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 18 T^{3} + \cdots + 1936)^{2} \) Copy content Toggle raw display
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