Properties

Label 480.2.f.e
Level $480$
Weight $2$
Character orbit 480.f
Analytic conductor $3.833$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,2,Mod(289,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.f (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: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.61803i
0.618034i
0.618034i
1.61803i
0 1.00000i 0 2.23607i 0 2.00000i 0 −1.00000 0
289.2 0 1.00000i 0 2.23607i 0 2.00000i 0 −1.00000 0
289.3 0 1.00000i 0 2.23607i 0 2.00000i 0 −1.00000 0
289.4 0 1.00000i 0 2.23607i 0 2.00000i 0 −1.00000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.f.e 4
3.b odd 2 1 1440.2.f.h 4
4.b odd 2 1 inner 480.2.f.e 4
5.b even 2 1 inner 480.2.f.e 4
5.c odd 4 1 2400.2.a.bi 2
5.c odd 4 1 2400.2.a.bj 2
8.b even 2 1 960.2.f.k 4
8.d odd 2 1 960.2.f.k 4
12.b even 2 1 1440.2.f.h 4
15.d odd 2 1 1440.2.f.h 4
15.e even 4 1 7200.2.a.cc 2
15.e even 4 1 7200.2.a.cq 2
16.e even 4 1 3840.2.d.bg 4
16.e even 4 1 3840.2.d.bh 4
16.f odd 4 1 3840.2.d.bg 4
16.f odd 4 1 3840.2.d.bh 4
20.d odd 2 1 inner 480.2.f.e 4
20.e even 4 1 2400.2.a.bi 2
20.e even 4 1 2400.2.a.bj 2
24.f even 2 1 2880.2.f.v 4
24.h odd 2 1 2880.2.f.v 4
40.e odd 2 1 960.2.f.k 4
40.f even 2 1 960.2.f.k 4
40.i odd 4 1 4800.2.a.cu 2
40.i odd 4 1 4800.2.a.cv 2
40.k even 4 1 4800.2.a.cu 2
40.k even 4 1 4800.2.a.cv 2
60.h even 2 1 1440.2.f.h 4
60.l odd 4 1 7200.2.a.cc 2
60.l odd 4 1 7200.2.a.cq 2
80.k odd 4 1 3840.2.d.bg 4
80.k odd 4 1 3840.2.d.bh 4
80.q even 4 1 3840.2.d.bg 4
80.q even 4 1 3840.2.d.bh 4
120.i odd 2 1 2880.2.f.v 4
120.m even 2 1 2880.2.f.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.e 4 1.a even 1 1 trivial
480.2.f.e 4 4.b odd 2 1 inner
480.2.f.e 4 5.b even 2 1 inner
480.2.f.e 4 20.d odd 2 1 inner
960.2.f.k 4 8.b even 2 1
960.2.f.k 4 8.d odd 2 1
960.2.f.k 4 40.e odd 2 1
960.2.f.k 4 40.f even 2 1
1440.2.f.h 4 3.b odd 2 1
1440.2.f.h 4 12.b even 2 1
1440.2.f.h 4 15.d odd 2 1
1440.2.f.h 4 60.h even 2 1
2400.2.a.bi 2 5.c odd 4 1
2400.2.a.bi 2 20.e even 4 1
2400.2.a.bj 2 5.c odd 4 1
2400.2.a.bj 2 20.e even 4 1
2880.2.f.v 4 24.f even 2 1
2880.2.f.v 4 24.h odd 2 1
2880.2.f.v 4 120.i odd 2 1
2880.2.f.v 4 120.m even 2 1
3840.2.d.bg 4 16.e even 4 1
3840.2.d.bg 4 16.f odd 4 1
3840.2.d.bg 4 80.k odd 4 1
3840.2.d.bg 4 80.q even 4 1
3840.2.d.bh 4 16.e even 4 1
3840.2.d.bh 4 16.f odd 4 1
3840.2.d.bh 4 80.k odd 4 1
3840.2.d.bh 4 80.q even 4 1
4800.2.a.cu 2 40.i odd 4 1
4800.2.a.cu 2 40.k even 4 1
4800.2.a.cv 2 40.i odd 4 1
4800.2.a.cv 2 40.k even 4 1
7200.2.a.cc 2 15.e even 4 1
7200.2.a.cc 2 60.l odd 4 1
7200.2.a.cq 2 15.e even 4 1
7200.2.a.cq 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$41$ \( (T - 10)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
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