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Properties

Label 520.1.bg.a
Level $520$
Weight $1$
Character orbit 520.bg
Analytic conductor $0.260$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
RM discriminant 104
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,1,Mod(77,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.77");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 520.bg (of order \(4\), 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: \(0.259513806569\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.104000.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{5} - \zeta_{8}^{3} q^{8} - \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{5} - \zeta_{8}^{3} q^{8} - \zeta_{8}^{2} q^{9} + \zeta_{8}^{2} q^{10} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{11} + \zeta_{8}^{3} q^{13} - q^{16} + ( - \zeta_{8}^{2} - 1) q^{17} + \zeta_{8}^{3} q^{18} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{19} - \zeta_{8}^{3} q^{20} + ( - \zeta_{8}^{2} - 1) q^{22} + ( - \zeta_{8}^{2} + 1) q^{23} + \zeta_{8}^{2} q^{25} + q^{26} + \zeta_{8} q^{32} + (\zeta_{8}^{3} + \zeta_{8}) q^{34} + q^{36} + (\zeta_{8}^{2} - 1) q^{38} - q^{40} + (\zeta_{8}^{3} + \zeta_{8}) q^{44} + \zeta_{8}^{3} q^{45} + (\zeta_{8}^{3} - \zeta_{8}) q^{46} + \zeta_{8}^{2} q^{49} - \zeta_{8}^{3} q^{50} - \zeta_{8} q^{52} + ( - \zeta_{8}^{2} - 1) q^{55} + (\zeta_{8}^{3} + \zeta_{8}) q^{59} - \zeta_{8}^{2} q^{64} + q^{65} + \zeta_{8} q^{67} + ( - \zeta_{8}^{2} + 1) q^{68} - \zeta_{8} q^{72} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{76} + \zeta_{8}^{2} q^{79} + \zeta_{8} q^{80} - q^{81} + (\zeta_{8}^{3} + \zeta_{8}) q^{85} + ( - \zeta_{8}^{2} + 1) q^{88} + q^{90} + (\zeta_{8}^{2} + 1) q^{92} + (\zeta_{8}^{2} - 1) q^{95} - \zeta_{8}^{3} q^{98} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{16} - 4 q^{17} - 4 q^{22} + 4 q^{23} + 4 q^{26} + 4 q^{36} - 4 q^{38} - 4 q^{40} - 4 q^{55} + 4 q^{65} + 4 q^{68} - 4 q^{81} + 4 q^{88} + 4 q^{90} + 4 q^{92} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
0.707107 + 0.707107i
−0.707107 − 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i
−0.707107 − 0.707107i 0 1.00000i −0.707107 − 0.707107i 0 0 0.707107 − 0.707107i − 1.00000i 1.00000i
77.2 0.707107 + 0.707107i 0 1.00000i 0.707107 + 0.707107i 0 0 −0.707107 + 0.707107i − 1.00000i 1.00000i
493.1 −0.707107 + 0.707107i 0 − 1.00000i −0.707107 + 0.707107i 0 0 0.707107 + 0.707107i 1.00000i − 1.00000i
493.2 0.707107 − 0.707107i 0 − 1.00000i 0.707107 − 0.707107i 0 0 −0.707107 − 0.707107i 1.00000i − 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.e even 2 1 RM by \(\Q(\sqrt{26}) \)
5.c odd 4 1 inner
8.b even 2 1 inner
13.b even 2 1 inner
40.i odd 4 1 inner
65.h odd 4 1 inner
520.bg odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.1.bg.a ✓ 4
4.b odd 2 1 2080.1.by.a 4
5.b even 2 1 2600.1.bg.a 4
5.c odd 4 1 inner 520.1.bg.a ✓ 4
5.c odd 4 1 2600.1.bg.a 4
8.b even 2 1 inner 520.1.bg.a ✓ 4
8.d odd 2 1 2080.1.by.a 4
13.b even 2 1 inner 520.1.bg.a ✓ 4
20.e even 4 1 2080.1.by.a 4
40.f even 2 1 2600.1.bg.a 4
40.i odd 4 1 inner 520.1.bg.a ✓ 4
40.i odd 4 1 2600.1.bg.a 4
40.k even 4 1 2080.1.by.a 4
52.b odd 2 1 2080.1.by.a 4
65.d even 2 1 2600.1.bg.a 4
65.h odd 4 1 inner 520.1.bg.a ✓ 4
65.h odd 4 1 2600.1.bg.a 4
104.e even 2 1 RM 520.1.bg.a ✓ 4
104.h odd 2 1 2080.1.by.a 4
260.p even 4 1 2080.1.by.a 4
520.p even 2 1 2600.1.bg.a 4
520.bc even 4 1 2080.1.by.a 4
520.bg odd 4 1 inner 520.1.bg.a ✓ 4
520.bg odd 4 1 2600.1.bg.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.1.bg.a ✓ 4 1.a even 1 1 trivial
520.1.bg.a ✓ 4 5.c odd 4 1 inner
520.1.bg.a ✓ 4 8.b even 2 1 inner
520.1.bg.a ✓ 4 13.b even 2 1 inner
520.1.bg.a ✓ 4 40.i odd 4 1 inner
520.1.bg.a ✓ 4 65.h odd 4 1 inner
520.1.bg.a ✓ 4 104.e even 2 1 RM
520.1.bg.a ✓ 4 520.bg odd 4 1 inner
2080.1.by.a 4 4.b odd 2 1
2080.1.by.a 4 8.d odd 2 1
2080.1.by.a 4 20.e even 4 1
2080.1.by.a 4 40.k even 4 1
2080.1.by.a 4 52.b odd 2 1
2080.1.by.a 4 104.h odd 2 1
2080.1.by.a 4 260.p even 4 1
2080.1.by.a 4 520.bc even 4 1
2600.1.bg.a 4 5.b even 2 1
2600.1.bg.a 4 5.c odd 4 1
2600.1.bg.a 4 40.f even 2 1
2600.1.bg.a 4 40.i odd 4 1
2600.1.bg.a 4 65.d even 2 1
2600.1.bg.a 4 65.h odd 4 1
2600.1.bg.a 4 520.p even 2 1
2600.1.bg.a 4 520.bg odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(520, [\chi])\).

Hecke characteristic polynomials

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