Properties

Label 8280.2.a.bb
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
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] = [8280,2,Mod(1,8280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8280.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.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: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + 2 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + 2 \beta q^{7} + 4 q^{11} + ( - \beta + 2) q^{13} + (2 \beta - 2) q^{17} + 4 q^{19} - q^{23} + q^{25} + (\beta + 6) q^{29} + ( - \beta + 4) q^{31} - 2 \beta q^{35} + ( - 2 \beta - 2) q^{37} + ( - \beta + 2) q^{41} + ( - 2 \beta - 4) q^{43} + (3 \beta - 4) q^{47} + (4 \beta + 9) q^{49} + (4 \beta - 6) q^{53} - 4 q^{55} + (4 \beta - 4) q^{59} + (2 \beta + 6) q^{61} + (\beta - 2) q^{65} + (4 \beta - 4) q^{67} + ( - 3 \beta + 4) q^{71} + (\beta + 14) q^{73} + 8 \beta q^{77} - 4 \beta q^{79} - 12 q^{83} + ( - 2 \beta + 2) q^{85} - 10 q^{89} + (2 \beta - 8) q^{91} - 4 q^{95} + ( - 4 \beta - 6) q^{97} +O(q^{100}) \) 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 \( 2 q - 2 q^{5} + 2 q^{7} + 8 q^{11} + 3 q^{13} - 2 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} + 13 q^{29} + 7 q^{31} - 2 q^{35} - 6 q^{37} + 3 q^{41} - 10 q^{43} - 5 q^{47} + 22 q^{49} - 8 q^{53} - 8 q^{55} - 4 q^{59} + 14 q^{61} - 3 q^{65} - 4 q^{67} + 5 q^{71} + 29 q^{73} + 8 q^{77} - 4 q^{79} - 24 q^{83} + 2 q^{85} - 20 q^{89} - 14 q^{91} - 8 q^{95} - 16 q^{97}+O(q^{100}) \) 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
−1.56155
2.56155
0 0 0 −1.00000 0 −3.12311 0 0 0
1.2 0 0 0 −1.00000 0 5.12311 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

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 8280.2.a.bb 2
3.b odd 2 1 920.2.a.f 2
12.b even 2 1 1840.2.a.k 2
15.d odd 2 1 4600.2.a.r 2
15.e even 4 2 4600.2.e.m 4
24.f even 2 1 7360.2.a.bm 2
24.h odd 2 1 7360.2.a.bj 2
60.h even 2 1 9200.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.f 2 3.b odd 2 1
1840.2.a.k 2 12.b even 2 1
4600.2.a.r 2 15.d odd 2 1
4600.2.e.m 4 15.e even 4 2
7360.2.a.bj 2 24.h odd 2 1
7360.2.a.bm 2 24.f even 2 1
8280.2.a.bb 2 1.a even 1 1 trivial
9200.2.a.bx 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{2} - 2T_{7} - 16 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} - 16 \) 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
$7$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$31$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$71$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 29T + 206 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 16T - 4 \) Copy content Toggle raw display
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