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Properties

Label 720.1.r.a
Level $720$
Weight $1$
Character orbit 720.r
Analytic conductor $0.359$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,1,Mod(19,720)]
 
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, 3, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 720.r (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: \(0.359326809096\)
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 4.0.92160.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{5} - \zeta_{8}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{5} - \zeta_{8}^{3} q^{8} - q^{10} - q^{16} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{17} + (\zeta_{8}^{2} + 1) q^{19} + \zeta_{8} q^{20} + (\zeta_{8}^{3} + \zeta_{8}) q^{23} - \zeta_{8}^{2} q^{25} - \zeta_{8}^{2} q^{31} + \zeta_{8} q^{32} + (\zeta_{8}^{2} - 1) q^{34} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{38} - \zeta_{8}^{2} q^{40} + ( - \zeta_{8}^{2} + 1) q^{46} + (\zeta_{8}^{3} - \zeta_{8}) q^{47} + q^{49} + \zeta_{8}^{3} q^{50} + (\zeta_{8}^{2} - 1) q^{61} + 2 \zeta_{8}^{3} q^{62} - \zeta_{8}^{2} q^{64} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{68} + (\zeta_{8}^{2} - 1) q^{76} + \zeta_{8}^{3} q^{80} + \zeta_{8}^{3} q^{83} + ( - \zeta_{8}^{2} - 1) q^{85} + (\zeta_{8}^{3} - \zeta_{8}) q^{92} + (\zeta_{8}^{2} + 1) q^{94} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{95} - \zeta_{8} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{10} - 4 q^{16} + 4 q^{19} - 4 q^{34} + 4 q^{46} + 4 q^{49} - 4 q^{61} - 4 q^{76} - 4 q^{85} + 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
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 0 −1.00000
19.2 0.707107 − 0.707107i 0 − 1.00000i −0.707107 − 0.707107i 0 0 −0.707107 − 0.707107i 0 −1.00000
379.1 −0.707107 − 0.707107i 0 1.00000i 0.707107 − 0.707107i 0 0 0.707107 − 0.707107i 0 −1.00000
379.2 0.707107 + 0.707107i 0 1.00000i −0.707107 + 0.707107i 0 0 −0.707107 + 0.707107i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.1.r.a ✓ 4
3.b odd 2 1 inner 720.1.r.a ✓ 4
4.b odd 2 1 2880.1.r.a 4
5.b even 2 1 inner 720.1.r.a ✓ 4
5.c odd 4 2 3600.1.bo.a 4
12.b even 2 1 2880.1.r.a 4
15.d odd 2 1 CM 720.1.r.a ✓ 4
15.e even 4 2 3600.1.bo.a 4
16.e even 4 1 2880.1.r.a 4
16.f odd 4 1 inner 720.1.r.a ✓ 4
20.d odd 2 1 2880.1.r.a 4
48.i odd 4 1 2880.1.r.a 4
48.k even 4 1 inner 720.1.r.a ✓ 4
60.h even 2 1 2880.1.r.a 4
80.j even 4 1 3600.1.bo.a 4
80.k odd 4 1 inner 720.1.r.a ✓ 4
80.q even 4 1 2880.1.r.a 4
80.s even 4 1 3600.1.bo.a 4
240.t even 4 1 inner 720.1.r.a ✓ 4
240.z odd 4 1 3600.1.bo.a 4
240.bd odd 4 1 3600.1.bo.a 4
240.bm odd 4 1 2880.1.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.r.a ✓ 4 1.a even 1 1 trivial
720.1.r.a ✓ 4 3.b odd 2 1 inner
720.1.r.a ✓ 4 5.b even 2 1 inner
720.1.r.a ✓ 4 15.d odd 2 1 CM
720.1.r.a ✓ 4 16.f odd 4 1 inner
720.1.r.a ✓ 4 48.k even 4 1 inner
720.1.r.a ✓ 4 80.k odd 4 1 inner
720.1.r.a ✓ 4 240.t even 4 1 inner
2880.1.r.a 4 4.b odd 2 1
2880.1.r.a 4 12.b even 2 1
2880.1.r.a 4 16.e even 4 1
2880.1.r.a 4 20.d odd 2 1
2880.1.r.a 4 48.i odd 4 1
2880.1.r.a 4 60.h even 2 1
2880.1.r.a 4 80.q even 4 1
2880.1.r.a 4 240.bm odd 4 1
3600.1.bo.a 4 5.c odd 4 2
3600.1.bo.a 4 15.e even 4 2
3600.1.bo.a 4 80.j even 4 1
3600.1.bo.a 4 80.s even 4 1
3600.1.bo.a 4 240.z odd 4 1
3600.1.bo.a 4 240.bd odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(720, [\chi])\).

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
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{2} \) 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
$47$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 16 \) 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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