Properties

Label 1040.2.f.e
Level $1040$
Weight $2$
Character orbit 1040.f
Analytic conductor $8.304$
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] = [1040,2,Mod(129,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.f (of order \(2\), degree \(1\), 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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12745506816.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{4} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + ( - \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{4} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + ( - \beta_1 - 2) q^{9} + \beta_{5} q^{11} + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - \beta_{5} - 5 \beta_{4} - 5 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{9} + 8 q^{25} - 24 q^{29} - 24 q^{35} - 24 q^{39} + 24 q^{49} + 32 q^{51} + 8 q^{55} - 8 q^{61} - 8 q^{69} + 48 q^{75} - 16 q^{79} + 80 q^{81} + 56 q^{91} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{4} + 16\nu^{2} + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 12\nu^{5} - 26\nu^{3} + 39\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} - 3\nu^{6} - 29\nu^{5} - 36\nu^{4} - 137\nu^{3} - 78\nu^{2} - 237\nu + 72 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} - 29\nu^{5} + 36\nu^{4} - 137\nu^{3} + 78\nu^{2} - 237\nu - 72 ) / 45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 76\nu^{3} - 87\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 76\nu^{3} - 105\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 9\nu^{6} - 29\nu^{5} + 108\nu^{4} - 137\nu^{3} + 324\nu^{2} - 237\nu + 144 ) / 45 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{4} + \beta_{3} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{6} - 6\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{7} + 16\beta_{4} - 8\beta_{3} + 3\beta _1 + 49 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -56\beta_{6} + 39\beta_{5} + 25\beta_{4} + 25\beta_{3} - 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 70\beta_{7} - 125\beta_{4} + 55\beta_{3} - 36\beta _1 - 332 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 451\beta_{6} - 273\beta_{5} - 248\beta_{4} - 248\beta_{3} + 4\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
0.281155i
2.81442i
1.65070i
2.29678i
1.65070i
2.29678i
0.281155i
2.81442i
0 3.09557i 0 −1.73205 1.41421i 0 4.37780 0 −6.58258 0
129.2 0 3.09557i 0 1.73205 + 1.41421i 0 −4.37780 0 −6.58258 0
129.3 0 0.646084i 0 −1.73205 + 1.41421i 0 −0.913701 0 2.58258 0
129.4 0 0.646084i 0 1.73205 1.41421i 0 0.913701 0 2.58258 0
129.5 0 0.646084i 0 −1.73205 1.41421i 0 −0.913701 0 2.58258 0
129.6 0 0.646084i 0 1.73205 + 1.41421i 0 0.913701 0 2.58258 0
129.7 0 3.09557i 0 −1.73205 + 1.41421i 0 4.37780 0 −6.58258 0
129.8 0 3.09557i 0 1.73205 1.41421i 0 −4.37780 0 −6.58258 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.f.e 8
4.b odd 2 1 260.2.d.a 8
5.b even 2 1 inner 1040.2.f.e 8
12.b even 2 1 2340.2.j.d 8
13.b even 2 1 inner 1040.2.f.e 8
20.d odd 2 1 260.2.d.a 8
20.e even 4 2 1300.2.f.f 8
52.b odd 2 1 260.2.d.a 8
52.f even 4 2 3380.2.c.d 8
60.h even 2 1 2340.2.j.d 8
65.d even 2 1 inner 1040.2.f.e 8
156.h even 2 1 2340.2.j.d 8
260.g odd 2 1 260.2.d.a 8
260.p even 4 2 1300.2.f.f 8
260.u even 4 2 3380.2.c.d 8
780.d even 2 1 2340.2.j.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.d.a 8 4.b odd 2 1
260.2.d.a 8 20.d odd 2 1
260.2.d.a 8 52.b odd 2 1
260.2.d.a 8 260.g odd 2 1
1040.2.f.e 8 1.a even 1 1 trivial
1040.2.f.e 8 5.b even 2 1 inner
1040.2.f.e 8 13.b even 2 1 inner
1040.2.f.e 8 65.d even 2 1 inner
1300.2.f.f 8 20.e even 4 2
1300.2.f.f 8 260.p even 4 2
2340.2.j.d 8 12.b even 2 1
2340.2.j.d 8 60.h even 2 1
2340.2.j.d 8 156.h even 2 1
2340.2.j.d 8 780.d even 2 1
3380.2.c.d 8 52.f even 4 2
3380.2.c.d 8 260.u even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{4} + 10T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 20T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{37}^{4} - 68T_{37}^{2} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 10 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 20 T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 22 T^{2} + 100)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 30 T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 34 T^{2} + 100)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 12)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 102 T^{2} + 900)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 68 T^{2} + 400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 88 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 66 T^{2} + 900)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 132 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 40 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 190 T^{2} + 8836)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 20)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 68 T^{2} + 400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 46 T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 68 T^{2} + 400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 80)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 276 T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 152 T^{2} + 400)^{2} \) Copy content Toggle raw display
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