Properties

Label 720.2.x.e
Level $720$
Weight $2$
Character orbit 720.x
Analytic conductor $5.749$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(127,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("720.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.x (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: \(5.74922894553\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 240)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 2) q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 2) q^{5} + \beta_1 q^{7} + ( - \beta_{3} - \beta_1) q^{11} + (3 \beta_{2} + 3) q^{13} + ( - \beta_{2} + 1) q^{17} + (\beta_{3} - \beta_1) q^{19} + \beta_{3} q^{23} + (4 \beta_{2} + 3) q^{25} + 4 \beta_{2} q^{29} + (\beta_{3} + 2 \beta_1) q^{35} + ( - 5 \beta_{2} + 5) q^{37} - \beta_{3} q^{43} - \beta_1 q^{47} + 9 \beta_{2} q^{49} + ( - \beta_{2} - 1) q^{53} + ( - 3 \beta_{3} - \beta_1) q^{55} + ( - 2 \beta_{3} + 2 \beta_1) q^{59} + 4 q^{61} + (9 \beta_{2} + 3) q^{65} - \beta_1 q^{67} + (\beta_{3} + \beta_1) q^{71} + ( - 3 \beta_{2} - 3) q^{73} + ( - 16 \beta_{2} + 16) q^{77} + (\beta_{3} - \beta_1) q^{79} + \beta_{3} q^{83} + ( - \beta_{2} + 3) q^{85} + 8 \beta_{2} q^{89} + (3 \beta_{3} + 3 \beta_1) q^{91} + (\beta_{3} - 3 \beta_1) q^{95} + (3 \beta_{2} - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} + 12 q^{13} + 4 q^{17} + 12 q^{25} + 20 q^{37} - 4 q^{53} + 16 q^{61} + 12 q^{65} - 12 q^{73} + 64 q^{77} + 12 q^{85} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 4\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 2.00000 1.00000i 0 −2.82843 + 2.82843i 0 0 0
127.2 0 0 0 2.00000 1.00000i 0 2.82843 2.82843i 0 0 0
703.1 0 0 0 2.00000 + 1.00000i 0 −2.82843 2.82843i 0 0 0
703.2 0 0 0 2.00000 + 1.00000i 0 2.82843 + 2.82843i 0 0 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 720.2.x.e 4
3.b odd 2 1 240.2.w.a 4
4.b odd 2 1 inner 720.2.x.e 4
5.b even 2 1 3600.2.x.d 4
5.c odd 4 1 inner 720.2.x.e 4
5.c odd 4 1 3600.2.x.d 4
12.b even 2 1 240.2.w.a 4
15.d odd 2 1 1200.2.w.a 4
15.e even 4 1 240.2.w.a 4
15.e even 4 1 1200.2.w.a 4
20.d odd 2 1 3600.2.x.d 4
20.e even 4 1 inner 720.2.x.e 4
20.e even 4 1 3600.2.x.d 4
24.f even 2 1 960.2.w.c 4
24.h odd 2 1 960.2.w.c 4
60.h even 2 1 1200.2.w.a 4
60.l odd 4 1 240.2.w.a 4
60.l odd 4 1 1200.2.w.a 4
120.q odd 4 1 960.2.w.c 4
120.w even 4 1 960.2.w.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.w.a 4 3.b odd 2 1
240.2.w.a 4 12.b even 2 1
240.2.w.a 4 15.e even 4 1
240.2.w.a 4 60.l odd 4 1
720.2.x.e 4 1.a even 1 1 trivial
720.2.x.e 4 4.b odd 2 1 inner
720.2.x.e 4 5.c odd 4 1 inner
720.2.x.e 4 20.e even 4 1 inner
960.2.w.c 4 24.f even 2 1
960.2.w.c 4 24.h odd 2 1
960.2.w.c 4 120.q odd 4 1
960.2.w.c 4 120.w even 4 1
1200.2.w.a 4 15.d odd 2 1
1200.2.w.a 4 15.e even 4 1
1200.2.w.a 4 60.h even 2 1
1200.2.w.a 4 60.l odd 4 1
3600.2.x.d 4 5.b even 2 1
3600.2.x.d 4 5.c odd 4 1
3600.2.x.d 4 20.d odd 2 1
3600.2.x.d 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{4} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} + 32 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 18 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
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