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Properties

Label 1560.2.bh.a
Level $1560$
Weight $2$
Character orbit 1560.bh
Analytic conductor $12.457$
Analytic rank $0$
Dimension $40$
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] = [1560,2,Mod(73,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.bh (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: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{5} - 4 q^{11} - 4 q^{13} - 20 q^{17} + 4 q^{19} + 8 q^{23} + 8 q^{25} - 4 q^{31} - 8 q^{33} - 4 q^{35} + 4 q^{39} - 20 q^{41} + 4 q^{45} - 80 q^{49} + 28 q^{53} - 4 q^{55} + 16 q^{57} - 20 q^{59} + 24 q^{61} + 8 q^{65} - 48 q^{67} + 24 q^{69} - 24 q^{71} - 48 q^{73} - 16 q^{75} + 32 q^{77} - 40 q^{81} + 48 q^{85} - 12 q^{87} - 44 q^{89} - 12 q^{91} - 72 q^{95} + 8 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 0 −0.707107 − 0.707107i 0 −2.16905 − 0.543325i 0 − 4.90149i 0 1.00000i 0
73.2 0 −0.707107 − 0.707107i 0 −1.92574 + 1.13645i 0 4.20892i 0 1.00000i 0
73.3 0 −0.707107 − 0.707107i 0 2.05414 − 0.883471i 0 3.23419i 0 1.00000i 0
73.4 0 −0.707107 − 0.707107i 0 1.42104 + 1.72645i 0 2.10790i 0 1.00000i 0
73.5 0 −0.707107 − 0.707107i 0 2.01064 − 0.978439i 0 2.05630i 0 1.00000i 0
73.6 0 −0.707107 − 0.707107i 0 −0.455384 − 2.18921i 0 1.27010i 0 1.00000i 0
73.7 0 −0.707107 − 0.707107i 0 −2.20592 + 0.365943i 0 − 1.38213i 0 1.00000i 0
73.8 0 −0.707107 − 0.707107i 0 −0.867701 − 2.06085i 0 − 0.395661i 0 1.00000i 0
73.9 0 −0.707107 − 0.707107i 0 2.20067 + 0.396312i 0 − 2.55892i 0 1.00000i 0
73.10 0 −0.707107 − 0.707107i 0 0.937318 + 2.03013i 0 − 3.63921i 0 1.00000i 0
73.11 0 0.707107 + 0.707107i 0 −0.270287 − 2.21967i 0 4.35926i 0 1.00000i 0
73.12 0 0.707107 + 0.707107i 0 −0.523765 − 2.17386i 0 − 4.72090i 0 1.00000i 0
73.13 0 0.707107 + 0.707107i 0 0.273518 + 2.21928i 0 − 2.48183i 0 1.00000i 0
73.14 0 0.707107 + 0.707107i 0 −1.71578 + 1.43392i 0 − 0.859681i 0 1.00000i 0
73.15 0 0.707107 + 0.707107i 0 2.21586 + 0.299925i 0 0.459071i 0 1.00000i 0
73.16 0 0.707107 + 0.707107i 0 1.12019 − 1.93524i 0 − 0.384898i 0 1.00000i 0
73.17 0 0.707107 + 0.707107i 0 −1.72772 − 1.41950i 0 1.20276i 0 1.00000i 0
73.18 0 0.707107 + 0.707107i 0 −1.74794 + 1.39452i 0 1.21119i 0 1.00000i 0
73.19 0 0.707107 + 0.707107i 0 2.18175 − 0.489859i 0 − 4.05390i 0 1.00000i 0
73.20 0 0.707107 + 0.707107i 0 1.19417 + 1.89049i 0 5.26893i 0 1.00000i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.20
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 1560.2.bh.a ✓ 40
5.c odd 4 1 1560.2.cy.a yes 40
13.d odd 4 1 1560.2.cy.a yes 40
65.k even 4 1 inner 1560.2.bh.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.bh.a ✓ 40 1.a even 1 1 trivial
1560.2.bh.a ✓ 40 65.k even 4 1 inner
1560.2.cy.a yes 40 5.c odd 4 1
1560.2.cy.a yes 40 13.d odd 4 1