Properties

Label 520.2.w.d
Level $520$
Weight $2$
Character orbit 520.w
Analytic conductor $4.152$
Analytic rank $0$
Dimension $4$
CM no
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(57,520)]
 
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, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.w (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: \(4.15222090511\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + (2 \beta_{2} - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + (2 \beta_{2} - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{9} + (\beta_{3} - \beta_{2} + 1) q^{11} + ( - 2 \beta_{2} + 3) q^{13} + (2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{15} + (\beta_{2} + 1) q^{17} + ( - 3 \beta_{3} - \beta_{2} + 1) q^{19} + ( - 2 \beta_{2} + 2) q^{21} + (\beta_{3} - \beta_{2} + 1) q^{23} + ( - 4 \beta_{2} - 3) q^{25} + (4 \beta_{3} - 8 \beta_{2} + 8) q^{27} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{29} + (3 \beta_{2} + \beta_1 + 3) q^{31} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{33} + ( - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{35} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 5) q^{39} + (\beta_{2} - 4 \beta_1 + 1) q^{41} + (\beta_{3} - 5 \beta_{2} + 5) q^{43} + ( - 2 \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 6) q^{45} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{47} + (4 \beta_{3} - 4 \beta_1 - 5) q^{49} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{51} + (\beta_{2} - 4 \beta_1 + 1) q^{53} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 1) q^{55} + (2 \beta_{3} - 2 \beta_1 + 10) q^{57} + (3 \beta_{2} - 3 \beta_1 + 3) q^{59} + (2 \beta_{3} - 2 \beta_1 - 2) q^{61} + (\beta_{3} - \beta_1 - 10) q^{63} + (8 \beta_{2} + 1) q^{65} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{67} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{69} + ( - 5 \beta_{2} - 3 \beta_1 - 5) q^{71} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{73} + ( - 4 \beta_{3} + 7 \beta_{2} - 3 \beta_1 - 1) q^{75} + (2 \beta_{2} + 2) q^{77} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{79} + ( - 6 \beta_{3} + 6 \beta_1 - 23) q^{81} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{83} + (\beta_{2} - 3) q^{85} + (4 \beta_{2} - 4) q^{87} + (9 \beta_{2} + 9) q^{89} + ( - \beta_{3} - 6 \beta_{2} - 5 \beta_1 - 4) q^{91} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{93} + (3 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 1) q^{95} + 10 q^{97} + (11 \beta_{2} - 7 \beta_1 + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{11} + 12 q^{13} + 12 q^{15} + 4 q^{17} + 4 q^{19} + 8 q^{21} + 4 q^{23} - 12 q^{25} + 32 q^{27} + 12 q^{31} - 24 q^{33} + 16 q^{35} - 20 q^{39} + 4 q^{41} + 20 q^{43} - 24 q^{45} - 20 q^{49} + 4 q^{53} + 4 q^{55} + 40 q^{57} + 12 q^{59} - 8 q^{61} - 40 q^{63} + 4 q^{65} + 16 q^{67} - 24 q^{69} - 20 q^{71} + 8 q^{73} - 4 q^{75} + 8 q^{77} - 92 q^{81} - 12 q^{85} - 16 q^{87} + 36 q^{89} - 16 q^{91} + 4 q^{95} + 40 q^{97} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 −2.41421 + 2.41421i 0 −1.00000 2.00000i 0 0.828427i 0 8.65685i 0
57.2 0 0.414214 0.414214i 0 −1.00000 2.00000i 0 4.82843i 0 2.65685i 0
73.1 0 −2.41421 2.41421i 0 −1.00000 + 2.00000i 0 0.828427i 0 8.65685i 0
73.2 0 0.414214 + 0.414214i 0 −1.00000 + 2.00000i 0 4.82843i 0 2.65685i 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
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.w.d 4
4.b odd 2 1 1040.2.bg.l 4
5.c odd 4 1 520.2.bh.d yes 4
13.d odd 4 1 520.2.bh.d yes 4
20.e even 4 1 1040.2.cd.l 4
52.f even 4 1 1040.2.cd.l 4
65.k even 4 1 inner 520.2.w.d 4
260.s odd 4 1 1040.2.bg.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.w.d 4 1.a even 1 1 trivial
520.2.w.d 4 65.k even 4 1 inner
520.2.bh.d yes 4 5.c odd 4 1
520.2.bh.d yes 4 13.d odd 4 1
1040.2.bg.l 4 4.b odd 2 1
1040.2.bg.l 4 260.s odd 4 1
1040.2.cd.l 4 20.e even 4 1
1040.2.cd.l 4 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 8 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 8 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + 8 T^{2} + 136 T + 1156 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + 8 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$29$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$37$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + 8 T^{2} + 248 T + 3844 \) Copy content Toggle raw display
$43$ \( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + 8 T^{2} + 248 T + 3844 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$83$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 162)^{2} \) Copy content Toggle raw display
$97$ \( (T - 10)^{4} \) Copy content Toggle raw display
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