Properties

Label 80.2.n.b
Level $80$
Weight $2$
Character orbit 80.n
Analytic conductor $0.639$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,2,Mod(47,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("80.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.n (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: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - 2 \beta_1 - 1) q^{5} + \beta_{3} q^{7} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - 2 \beta_1 - 1) q^{5} + \beta_{3} q^{7} + 3 \beta_1 q^{9} + ( - \beta_{3} + \beta_{2}) q^{11} + ( - \beta_1 + 1) q^{13} + ( - 2 \beta_{3} + \beta_{2}) q^{15} + (\beta_1 + 1) q^{17} + (2 \beta_{3} + 2 \beta_{2}) q^{19} - 6 q^{21} - \beta_{2} q^{23} + (4 \beta_1 - 3) q^{25} - 4 \beta_1 q^{29} + (\beta_{3} - \beta_{2}) q^{31} + ( - 6 \beta_1 + 6) q^{33} + ( - \beta_{3} - 2 \beta_{2}) q^{35} + (5 \beta_1 + 5) q^{37} + ( - \beta_{3} - \beta_{2}) q^{39} + 2 q^{41} - \beta_{2} q^{43} + ( - 3 \beta_1 + 6) q^{45} + \beta_{3} q^{47} + \beta_1 q^{49} + (\beta_{3} - \beta_{2}) q^{51} + (7 \beta_1 - 7) q^{53} + (3 \beta_{3} + \beta_{2}) q^{55} + ( - 12 \beta_1 - 12) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{59} + 6 q^{61} + 3 \beta_{2} q^{63} + ( - \beta_1 - 3) q^{65} - 3 \beta_{3} q^{67} + 6 \beta_1 q^{69} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{71} + (7 \beta_1 - 7) q^{73} + (4 \beta_{3} + 3 \beta_{2}) q^{75} + (6 \beta_1 + 6) q^{77} + 9 q^{81} + 7 \beta_{2} q^{83} + ( - 3 \beta_1 + 1) q^{85} - 4 \beta_{3} q^{87} - 8 \beta_1 q^{89} + (\beta_{3} - \beta_{2}) q^{91} + (6 \beta_1 - 6) q^{93} + (2 \beta_{3} - 6 \beta_{2}) q^{95} + ( - 7 \beta_1 - 7) q^{97} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 4 q^{13} + 4 q^{17} - 24 q^{21} - 12 q^{25} + 24 q^{33} + 20 q^{37} + 8 q^{41} + 24 q^{45} - 28 q^{53} - 48 q^{57} + 24 q^{61} - 12 q^{65} - 28 q^{73} + 24 q^{77} + 36 q^{81} + 4 q^{85} - 24 q^{93} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} + 2\zeta_{12} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} - 2\zeta_{12}^{2} + 2\zeta_{12} + 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0 −1.73205 1.73205i 0 −1.00000 2.00000i 0 1.73205 1.73205i 0 3.00000i 0
47.2 0 1.73205 + 1.73205i 0 −1.00000 2.00000i 0 −1.73205 + 1.73205i 0 3.00000i 0
63.1 0 −1.73205 + 1.73205i 0 −1.00000 + 2.00000i 0 1.73205 + 1.73205i 0 3.00000i 0
63.2 0 1.73205 1.73205i 0 −1.00000 + 2.00000i 0 −1.73205 1.73205i 0 3.00000i 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.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.2.n.b 4
3.b odd 2 1 720.2.x.d 4
4.b odd 2 1 inner 80.2.n.b 4
5.b even 2 1 400.2.n.b 4
5.c odd 4 1 inner 80.2.n.b 4
5.c odd 4 1 400.2.n.b 4
8.b even 2 1 320.2.n.i 4
8.d odd 2 1 320.2.n.i 4
12.b even 2 1 720.2.x.d 4
15.d odd 2 1 3600.2.x.e 4
15.e even 4 1 720.2.x.d 4
15.e even 4 1 3600.2.x.e 4
16.e even 4 1 1280.2.o.q 4
16.e even 4 1 1280.2.o.r 4
16.f odd 4 1 1280.2.o.q 4
16.f odd 4 1 1280.2.o.r 4
20.d odd 2 1 400.2.n.b 4
20.e even 4 1 inner 80.2.n.b 4
20.e even 4 1 400.2.n.b 4
40.e odd 2 1 1600.2.n.r 4
40.f even 2 1 1600.2.n.r 4
40.i odd 4 1 320.2.n.i 4
40.i odd 4 1 1600.2.n.r 4
40.k even 4 1 320.2.n.i 4
40.k even 4 1 1600.2.n.r 4
60.h even 2 1 3600.2.x.e 4
60.l odd 4 1 720.2.x.d 4
60.l odd 4 1 3600.2.x.e 4
80.i odd 4 1 1280.2.o.r 4
80.j even 4 1 1280.2.o.q 4
80.s even 4 1 1280.2.o.r 4
80.t odd 4 1 1280.2.o.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.b 4 1.a even 1 1 trivial
80.2.n.b 4 4.b odd 2 1 inner
80.2.n.b 4 5.c odd 4 1 inner
80.2.n.b 4 20.e even 4 1 inner
320.2.n.i 4 8.b even 2 1
320.2.n.i 4 8.d odd 2 1
320.2.n.i 4 40.i odd 4 1
320.2.n.i 4 40.k even 4 1
400.2.n.b 4 5.b even 2 1
400.2.n.b 4 5.c odd 4 1
400.2.n.b 4 20.d odd 2 1
400.2.n.b 4 20.e even 4 1
720.2.x.d 4 3.b odd 2 1
720.2.x.d 4 12.b even 2 1
720.2.x.d 4 15.e even 4 1
720.2.x.d 4 60.l odd 4 1
1280.2.o.q 4 16.e even 4 1
1280.2.o.q 4 16.f odd 4 1
1280.2.o.q 4 80.j even 4 1
1280.2.o.q 4 80.t odd 4 1
1280.2.o.r 4 16.e even 4 1
1280.2.o.r 4 16.f odd 4 1
1280.2.o.r 4 80.i odd 4 1
1280.2.o.r 4 80.s even 4 1
1600.2.n.r 4 40.e odd 2 1
1600.2.n.r 4 40.f even 2 1
1600.2.n.r 4 40.i odd 4 1
1600.2.n.r 4 40.k even 4 1
3600.2.x.e 4 15.d odd 2 1
3600.2.x.e 4 15.e even 4 1
3600.2.x.e 4 60.h even 2 1
3600.2.x.e 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 36 \) acting on \(S_{2}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 36 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 36 \) Copy content Toggle raw display
$47$ \( T^{4} + 36 \) Copy content Toggle raw display
$53$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 2916 \) Copy content Toggle raw display
$71$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 86436 \) Copy content Toggle raw display
$89$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
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