Properties

Label 520.2.d.a
Level $520$
Weight $2$
Character orbit 520.d
Analytic conductor $4.152$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(209,520)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 2) q^{5} - 4 i q^{7} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i + 2) q^{5} - 4 i q^{7} + 3 q^{9} - 2 q^{11} + i q^{13} - 4 i q^{17} + 2 q^{19} - 2 i q^{23} + (4 i + 3) q^{25} - 6 q^{29} + 8 q^{31} + ( - 8 i + 4) q^{35} + 6 i q^{37} + 10 q^{41} + 4 i q^{43} + (3 i + 6) q^{45} - 9 q^{49} - 6 i q^{53} + ( - 2 i - 4) q^{55} - 6 q^{59} + 2 q^{61} - 12 i q^{63} + (2 i - 1) q^{65} - 4 i q^{67} - 12 q^{71} + 2 i q^{73} + 8 i q^{77} - 8 q^{79} + 9 q^{81} + 12 i q^{83} + ( - 8 i + 4) q^{85} - 14 q^{89} + 4 q^{91} + (2 i + 4) q^{95} + 10 i q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{25} - 12 q^{29} + 16 q^{31} + 8 q^{35} + 20 q^{41} + 12 q^{45} - 18 q^{49} - 8 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 24 q^{71} - 16 q^{79} + 18 q^{81} + 8 q^{85} - 28 q^{89} + 8 q^{91} + 8 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
1.00000i
1.00000i
0 0 0 2.00000 1.00000i 0 4.00000i 0 3.00000 0
209.2 0 0 0 2.00000 + 1.00000i 0 4.00000i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.d.a 2
3.b odd 2 1 4680.2.l.a 2
4.b odd 2 1 1040.2.d.a 2
5.b even 2 1 inner 520.2.d.a 2
5.c odd 4 1 2600.2.a.g 1
5.c odd 4 1 2600.2.a.i 1
15.d odd 2 1 4680.2.l.a 2
20.d odd 2 1 1040.2.d.a 2
20.e even 4 1 5200.2.a.o 1
20.e even 4 1 5200.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.d.a 2 1.a even 1 1 trivial
520.2.d.a 2 5.b even 2 1 inner
1040.2.d.a 2 4.b odd 2 1
1040.2.d.a 2 20.d odd 2 1
2600.2.a.g 1 5.c odd 4 1
2600.2.a.i 1 5.c odd 4 1
4680.2.l.a 2 3.b odd 2 1
4680.2.l.a 2 15.d odd 2 1
5200.2.a.o 1 20.e even 4 1
5200.2.a.u 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). 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
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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