Properties

Label 520.2.a.c
Level $520$
Weight $2$
Character orbit 520.a
Self dual yes
Analytic conductor $4.152$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(1,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, 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("520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.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: \(4.15222090511\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta - 2) q^{3} + q^{5} - 2 q^{7} + ( - 4 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 2) q^{3} + q^{5} - 2 q^{7} + ( - 4 \beta + 3) q^{9} - 3 \beta q^{11} - q^{13} + (\beta - 2) q^{15} + (2 \beta - 2) q^{17} + (3 \beta - 4) q^{19} + ( - 2 \beta + 4) q^{21} + ( - 5 \beta - 2) q^{23} + q^{25} + (8 \beta - 8) q^{27} + (4 \beta - 4) q^{29} + \beta q^{31} + (6 \beta - 6) q^{33} - 2 q^{35} + ( - 2 \beta - 4) q^{37} + ( - \beta + 2) q^{39} + (2 \beta + 2) q^{41} + ( - 3 \beta - 6) q^{43} + ( - 4 \beta + 3) q^{45} + 2 q^{47} - 3 q^{49} + ( - 6 \beta + 8) q^{51} + ( - 2 \beta - 6) q^{53} - 3 \beta q^{55} + ( - 10 \beta + 14) q^{57} + (\beta - 4) q^{59} + (8 \beta + 4) q^{61} + (8 \beta - 6) q^{63} - q^{65} + ( - 2 \beta - 2) q^{67} + (8 \beta - 6) q^{69} + 7 \beta q^{71} + ( - 2 \beta + 4) q^{73} + (\beta - 2) q^{75} + 6 \beta q^{77} + ( - 2 \beta + 4) q^{79} + ( - 12 \beta + 23) q^{81} - 2 q^{83} + (2 \beta - 2) q^{85} + ( - 12 \beta + 16) q^{87} - 10 q^{89} + 2 q^{91} + ( - 2 \beta + 2) q^{93} + (3 \beta - 4) q^{95} + (4 \beta + 6) q^{97} + ( - 9 \beta + 24) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} - 2 q^{13} - 4 q^{15} - 4 q^{17} - 8 q^{19} + 8 q^{21} - 4 q^{23} + 2 q^{25} - 16 q^{27} - 8 q^{29} - 12 q^{33} - 4 q^{35} - 8 q^{37} + 4 q^{39} + 4 q^{41} - 12 q^{43} + 6 q^{45} + 4 q^{47} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 28 q^{57} - 8 q^{59} + 8 q^{61} - 12 q^{63} - 2 q^{65} - 4 q^{67} - 12 q^{69} + 8 q^{73} - 4 q^{75} + 8 q^{79} + 46 q^{81} - 4 q^{83} - 4 q^{85} + 32 q^{87} - 20 q^{89} + 4 q^{91} + 4 q^{93} - 8 q^{95} + 12 q^{97} + 48 q^{99}+O(q^{100}) \) 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 −3.41421 0 1.00000 0 −2.00000 0 8.65685 0
1.2 0 −0.585786 0 1.00000 0 −2.00000 0 −2.65685 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(13\) \( +1 \)

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 520.2.a.c 2
3.b odd 2 1 4680.2.a.w 2
4.b odd 2 1 1040.2.a.n 2
5.b even 2 1 2600.2.a.w 2
5.c odd 4 2 2600.2.d.i 4
8.b even 2 1 4160.2.a.bn 2
8.d odd 2 1 4160.2.a.u 2
12.b even 2 1 9360.2.a.ck 2
13.b even 2 1 6760.2.a.n 2
20.d odd 2 1 5200.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.c 2 1.a even 1 1 trivial
1040.2.a.n 2 4.b odd 2 1
2600.2.a.w 2 5.b even 2 1
2600.2.d.i 4 5.c odd 4 2
4160.2.a.u 2 8.d odd 2 1
4160.2.a.bn 2 8.b even 2 1
4680.2.a.w 2 3.b odd 2 1
5200.2.a.bl 2 20.d odd 2 1
6760.2.a.n 2 13.b even 2 1
9360.2.a.ck 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(520))\):

\( T_{3}^{2} + 4T_{3} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 18 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T - 2 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 46 \) Copy content Toggle raw display
$29$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$31$ \( T^{2} - 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 98 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$83$ \( (T + 2)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
show more
show less