Properties

Label 8640.2.a.cy
Level $8640$
Weight $2$
Character orbit 8640.a
Self dual yes
Analytic conductor $68.991$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8640,2,Mod(1,8640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8640 = 2^{6} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8640.a (trivial)

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: yes
Analytic conductor: \(68.9907473464\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + ( - \beta - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + ( - \beta - 1) q^{7} + (\beta - 1) q^{11} + (\beta - 3) q^{13} + (\beta + 2) q^{17} - \beta q^{19} - 3 q^{23} + q^{25} + (\beta + 5) q^{29} + ( - \beta + 2) q^{31} + ( - \beta - 1) q^{35} - 2 q^{37} + ( - \beta + 1) q^{41} + ( - \beta - 3) q^{43} + (2 \beta - 2) q^{47} + (2 \beta + 7) q^{49} + ( - \beta - 2) q^{53} + (\beta - 1) q^{55} + (\beta + 5) q^{59} + ( - 2 \beta - 3) q^{61} + (\beta - 3) q^{65} + ( - 2 \beta - 8) q^{67} + (\beta + 11) q^{71} + ( - \beta + 9) q^{73} - 12 q^{77} + ( - \beta + 8) q^{79} - 3 q^{83} + (\beta + 2) q^{85} + ( - 3 \beta + 3) q^{89} + (2 \beta - 10) q^{91} - \beta q^{95} + 8 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} - 6 q^{13} + 4 q^{17} - 6 q^{23} + 2 q^{25} + 10 q^{29} + 4 q^{31} - 2 q^{35} - 4 q^{37} + 2 q^{41} - 6 q^{43} - 4 q^{47} + 14 q^{49} - 4 q^{53} - 2 q^{55} + 10 q^{59} - 6 q^{61} - 6 q^{65} - 16 q^{67} + 22 q^{71} + 18 q^{73} - 24 q^{77} + 16 q^{79} - 6 q^{83} + 4 q^{85} + 6 q^{89} - 20 q^{91} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.30278
−1.30278
0 0 0 1.00000 0 −4.60555 0 0 0
1.2 0 0 0 1.00000 0 2.60555 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8640.2.a.cy 2
3.b odd 2 1 8640.2.a.ck 2
4.b odd 2 1 8640.2.a.df 2
8.b even 2 1 2160.2.a.y 2
8.d odd 2 1 135.2.a.d yes 2
12.b even 2 1 8640.2.a.cr 2
24.f even 2 1 135.2.a.c 2
24.h odd 2 1 2160.2.a.ba 2
40.e odd 2 1 675.2.a.k 2
40.k even 4 2 675.2.b.h 4
56.e even 2 1 6615.2.a.v 2
72.l even 6 2 405.2.e.k 4
72.p odd 6 2 405.2.e.j 4
120.m even 2 1 675.2.a.p 2
120.q odd 4 2 675.2.b.i 4
168.e odd 2 1 6615.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c 2 24.f even 2 1
135.2.a.d yes 2 8.d odd 2 1
405.2.e.j 4 72.p odd 6 2
405.2.e.k 4 72.l even 6 2
675.2.a.k 2 40.e odd 2 1
675.2.a.p 2 120.m even 2 1
675.2.b.h 4 40.k even 4 2
675.2.b.i 4 120.q odd 4 2
2160.2.a.y 2 8.b even 2 1
2160.2.a.ba 2 24.h odd 2 1
6615.2.a.p 2 168.e odd 2 1
6615.2.a.v 2 56.e even 2 1
8640.2.a.ck 2 3.b odd 2 1
8640.2.a.cr 2 12.b even 2 1
8640.2.a.cy 2 1.a even 1 1 trivial
8640.2.a.df 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8640))\):

\( T_{7}^{2} + 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 9 \) Copy content Toggle raw display
\( T_{19}^{2} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 13 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 43 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 22T + 108 \) Copy content Toggle raw display
$73$ \( T^{2} - 18T + 68 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 51 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 108 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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