Properties

Label 4560.2.a.bf
Level $4560$
Weight $2$
Character orbit 4560.a
Self dual yes
Analytic conductor $36.412$
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] = [4560,2,Mod(1,4560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4560, 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("4560.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.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: \(36.4117833217\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + \beta q^{7} + q^{9} + (3 \beta - 2) q^{11} + ( - \beta - 2) q^{13} + q^{15} + ( - 2 \beta + 4) q^{17} + q^{19} - \beta q^{21} + ( - 4 \beta - 2) q^{23} + q^{25} - q^{27} + \beta q^{29} + ( - 2 \beta + 6) q^{31} + ( - 3 \beta + 2) q^{33} - \beta q^{35} + ( - \beta - 2) q^{37} + (\beta + 2) q^{39} + ( - 3 \beta + 4) q^{41} + ( - 3 \beta - 8) q^{43} - q^{45} + (4 \beta + 2) q^{47} - 5 q^{49} + (2 \beta - 4) q^{51} + 8 q^{53} + ( - 3 \beta + 2) q^{55} - q^{57} + ( - 6 \beta - 4) q^{59} + (8 \beta - 4) q^{61} + \beta q^{63} + (\beta + 2) q^{65} + (4 \beta + 4) q^{67} + (4 \beta + 2) q^{69} + (2 \beta + 8) q^{71} + ( - 4 \beta - 2) q^{73} - q^{75} + ( - 2 \beta + 6) q^{77} + q^{81} + ( - 2 \beta - 10) q^{83} + (2 \beta - 4) q^{85} - \beta q^{87} + (7 \beta - 4) q^{89} + ( - 2 \beta - 2) q^{91} + (2 \beta - 6) q^{93} - q^{95} + (3 \beta - 14) q^{97} + (3 \beta - 2) q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} - 4 q^{13} + 2 q^{15} + 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{31} + 4 q^{33} - 4 q^{37} + 4 q^{39} + 8 q^{41} - 16 q^{43} - 2 q^{45} + 4 q^{47} - 10 q^{49} - 8 q^{51} + 16 q^{53} + 4 q^{55} - 2 q^{57} - 8 q^{59} - 8 q^{61} + 4 q^{65} + 8 q^{67} + 4 q^{69} + 16 q^{71} - 4 q^{73} - 2 q^{75} + 12 q^{77} + 2 q^{81} - 20 q^{83} - 8 q^{85} - 8 q^{89} - 4 q^{91} - 12 q^{93} - 2 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 −1.00000 0 −1.00000 0 −1.41421 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.41421 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(19\) \(-1\)

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 4560.2.a.bf 2
4.b odd 2 1 285.2.a.g 2
12.b even 2 1 855.2.a.d 2
20.d odd 2 1 1425.2.a.k 2
20.e even 4 2 1425.2.c.l 4
60.h even 2 1 4275.2.a.y 2
76.d even 2 1 5415.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.g 2 4.b odd 2 1
855.2.a.d 2 12.b even 2 1
1425.2.a.k 2 20.d odd 2 1
1425.2.c.l 4 20.e even 4 2
4275.2.a.y 2 60.h even 2 1
4560.2.a.bf 2 1.a even 1 1 trivial
5415.2.a.n 2 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4560))\):

\( T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 14 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 8T_{17} + 8 \) 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
$7$ \( T^{2} - 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$29$ \( T^{2} - 2 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 16T + 46 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$53$ \( (T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 92 \) Copy content Toggle raw display
$89$ \( T^{2} + 8T - 82 \) Copy content Toggle raw display
$97$ \( T^{2} + 28T + 178 \) Copy content Toggle raw display
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