Properties

Label 8880.2.a.bn
Level $8880$
Weight $2$
Character orbit 8880.a
Self dual yes
Analytic conductor $70.907$
Analytic rank $1$
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] = [8880,2,Mod(1,8880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8880, 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("8880.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8880.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: \(70.9071569949\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( - 2 \beta + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + ( - 2 \beta + 1) q^{7} + q^{9} - \beta q^{11} + ( - 2 \beta - 1) q^{13} - q^{15} + (3 \beta - 3) q^{17} + ( - \beta + 7) q^{19} + ( - 2 \beta + 1) q^{21} + (4 \beta - 5) q^{23} + q^{25} + q^{27} + (3 \beta - 7) q^{29} + (4 \beta + 3) q^{31} - \beta q^{33} + (2 \beta - 1) q^{35} - q^{37} + ( - 2 \beta - 1) q^{39} + ( - 4 \beta - 1) q^{41} + (9 \beta - 7) q^{43} - q^{45} + (2 \beta - 7) q^{47} - 2 q^{49} + (3 \beta - 3) q^{51} + 4 \beta q^{53} + \beta q^{55} + ( - \beta + 7) q^{57} + (7 \beta - 6) q^{59} + (4 \beta - 11) q^{61} + ( - 2 \beta + 1) q^{63} + (2 \beta + 1) q^{65} + ( - 7 \beta + 5) q^{67} + (4 \beta - 5) q^{69} + ( - \beta - 7) q^{71} + ( - 9 \beta + 8) q^{73} + q^{75} + (\beta + 2) q^{77} + (\beta + 4) q^{79} + q^{81} + ( - 9 \beta + 10) q^{83} + ( - 3 \beta + 3) q^{85} + (3 \beta - 7) q^{87} + (8 \beta - 13) q^{89} + (4 \beta + 3) q^{91} + (4 \beta + 3) q^{93} + (\beta - 7) q^{95} + (4 \beta + 1) q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - q^{11} - 4 q^{13} - 2 q^{15} - 3 q^{17} + 13 q^{19} - 6 q^{23} + 2 q^{25} + 2 q^{27} - 11 q^{29} + 10 q^{31} - q^{33} - 2 q^{37} - 4 q^{39} - 6 q^{41} - 5 q^{43} - 2 q^{45} - 12 q^{47} - 4 q^{49} - 3 q^{51} + 4 q^{53} + q^{55} + 13 q^{57} - 5 q^{59} - 18 q^{61} + 4 q^{65} + 3 q^{67} - 6 q^{69} - 15 q^{71} + 7 q^{73} + 2 q^{75} + 5 q^{77} + 9 q^{79} + 2 q^{81} + 11 q^{83} + 3 q^{85} - 11 q^{87} - 18 q^{89} + 10 q^{91} + 10 q^{93} - 13 q^{95} + 6 q^{97} - q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 1.00000 0 −1.00000 0 −2.23607 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.23607 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(37\) \(1\)

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 8880.2.a.bn 2
4.b odd 2 1 555.2.a.f 2
12.b even 2 1 1665.2.a.h 2
20.d odd 2 1 2775.2.a.m 2
60.h even 2 1 8325.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.a.f 2 4.b odd 2 1
1665.2.a.h 2 12.b even 2 1
2775.2.a.m 2 20.d odd 2 1
8325.2.a.bl 2 60.h even 2 1
8880.2.a.bn 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8880))\):

\( T_{7}^{2} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 1 \) Copy content Toggle raw display
\( T_{23}^{2} + 6T_{23} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5 \) Copy content Toggle raw display
$11$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} + 11T + 19 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 5 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T - 95 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 55 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 61 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 55 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T - 89 \) Copy content Toggle raw display
$79$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$83$ \( T^{2} - 11T - 71 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
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