Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(81,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.q (of order \(3\), degree \(2\), 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: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} - q^{5} + 3 \zeta_{6} q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} - q^{5} + 3 \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} + ( - 4 \zeta_{6} + 3) q^{13} + (\zeta_{6} - 1) q^{15} + 3 \zeta_{6} q^{17} + 5 \zeta_{6} q^{19} + 3 q^{21} + (3 \zeta_{6} - 3) q^{23} + q^{25} + 5 q^{27} + ( - 5 \zeta_{6} + 5) q^{29} - 3 \zeta_{6} q^{33} - 3 \zeta_{6} q^{35} + ( - 11 \zeta_{6} + 11) q^{37} + ( - 3 \zeta_{6} - 1) q^{39} + ( - 5 \zeta_{6} + 5) q^{41} + 7 \zeta_{6} q^{43} - 2 \zeta_{6} q^{45} - 8 q^{47} + (2 \zeta_{6} - 2) q^{49} + 3 q^{51} - 10 q^{53} + (3 \zeta_{6} - 3) q^{55} + 5 q^{57} + \zeta_{6} q^{59} + 5 \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{63} + (4 \zeta_{6} - 3) q^{65} + (11 \zeta_{6} - 11) q^{67} + 3 \zeta_{6} q^{69} - 3 \zeta_{6} q^{71} - 6 q^{73} + ( - \zeta_{6} + 1) q^{75} + 9 q^{77} + (\zeta_{6} - 1) q^{81} - 16 q^{83} - 3 \zeta_{6} q^{85} - 5 \zeta_{6} q^{87} + ( - 13 \zeta_{6} + 13) q^{89} + ( - 3 \zeta_{6} + 12) q^{91} - 5 \zeta_{6} q^{95} + 7 \zeta_{6} q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{5} + 3 q^{7} + 2 q^{9} + 3 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} + 5 q^{19} + 6 q^{21} - 3 q^{23} + 2 q^{25} + 10 q^{27} + 5 q^{29} - 3 q^{33} - 3 q^{35} + 11 q^{37} - 5 q^{39} + 5 q^{41} + 7 q^{43} - 2 q^{45} - 16 q^{47} - 2 q^{49} + 6 q^{51} - 20 q^{53} - 3 q^{55} + 10 q^{57} + q^{59} + 5 q^{61} - 6 q^{63} - 2 q^{65} - 11 q^{67} + 3 q^{69} - 3 q^{71} - 12 q^{73} + q^{75} + 18 q^{77} - q^{81} - 32 q^{83} - 3 q^{85} - 5 q^{87} + 13 q^{89} + 21 q^{91} - 5 q^{95} + 7 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 −1.00000 0 1.50000 2.59808i 0 1.00000 1.73205i 0
321.1 0 0.500000 0.866025i 0 −1.00000 0 1.50000 + 2.59808i 0 1.00000 + 1.73205i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.q.c 2
4.b odd 2 1 1040.2.q.d 2
13.c even 3 1 inner 520.2.q.c 2
13.c even 3 1 6760.2.a.c 1
13.e even 6 1 6760.2.a.h 1
52.j odd 6 1 1040.2.q.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.q.c 2 1.a even 1 1 trivial
520.2.q.c 2 13.c even 3 1 inner
1040.2.q.d 2 4.b odd 2 1
1040.2.q.d 2 52.j odd 6 1
6760.2.a.c 1 13.c even 3 1
6760.2.a.h 1 13.e even 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$97$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
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