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Properties

Label 1560.2.cy.b
Level $1560$
Weight $2$
Character orbit 1560.cy
Analytic conductor $12.457$
Analytic rank $0$
Dimension $44$
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(697,1560)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.697");
 
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.cy (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: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{11} - 4 q^{13} - 8 q^{17} - 4 q^{19} + 8 q^{23} + 12 q^{25} - 4 q^{31} + 4 q^{35} + 12 q^{39} - 24 q^{41} + 4 q^{45} + 8 q^{47} + 52 q^{49} - 16 q^{53} + 52 q^{55} - 12 q^{59} - 24 q^{61} - 20 q^{65} + 24 q^{69} + 24 q^{71} - 16 q^{75} - 16 q^{77} - 44 q^{81} + 8 q^{83} - 44 q^{85} + 12 q^{87} - 40 q^{89} + 12 q^{91} - 8 q^{95} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
697.1 0 −0.707107 + 0.707107i 0 −1.51815 − 1.64171i 0 3.69312 0 − 1.00000i 0
697.2 0 −0.707107 + 0.707107i 0 1.98070 + 1.03771i 0 3.64284 0 − 1.00000i 0
697.3 0 −0.707107 + 0.707107i 0 1.77198 − 1.36385i 0 2.31001 0 − 1.00000i 0
697.4 0 −0.707107 + 0.707107i 0 −2.08690 + 0.803019i 0 1.54723 0 − 1.00000i 0
697.5 0 −0.707107 + 0.707107i 0 −1.58530 + 1.57697i 0 1.86435 0 − 1.00000i 0
697.6 0 −0.707107 + 0.707107i 0 0.988709 − 2.00561i 0 −1.09144 0 − 1.00000i 0
697.7 0 −0.707107 + 0.707107i 0 0.0788104 + 2.23468i 0 2.19599 0 − 1.00000i 0
697.8 0 −0.707107 + 0.707107i 0 2.22552 + 0.216930i 0 −2.74283 0 − 1.00000i 0
697.9 0 −0.707107 + 0.707107i 0 1.46413 + 1.69007i 0 −3.50399 0 − 1.00000i 0
697.10 0 −0.707107 + 0.707107i 0 −1.11746 − 1.93682i 0 −3.50978 0 − 1.00000i 0
697.11 0 −0.707107 + 0.707107i 0 −2.20204 + 0.388622i 0 −4.40548 0 − 1.00000i 0
697.12 0 0.707107 − 0.707107i 0 −0.269401 + 2.21978i 0 5.27208 0 − 1.00000i 0
697.13 0 0.707107 − 0.707107i 0 0.702268 − 2.12293i 0 3.49765 0 − 1.00000i 0
697.14 0 0.707107 − 0.707107i 0 2.16360 + 0.564644i 0 2.24732 0 − 1.00000i 0
697.15 0 0.707107 − 0.707107i 0 2.20208 − 0.388368i 0 0.876923 0 − 1.00000i 0
697.16 0 0.707107 − 0.707107i 0 −2.04927 + 0.894711i 0 0.0953925 0 − 1.00000i 0
697.17 0 0.707107 − 0.707107i 0 −2.21088 + 0.334653i 0 0.504934 0 − 1.00000i 0
697.18 0 0.707107 − 0.707107i 0 −1.44422 − 1.70711i 0 −0.594486 0 − 1.00000i 0
697.19 0 0.707107 − 0.707107i 0 −0.848058 − 2.06901i 0 −1.95557 0 − 1.00000i 0
697.20 0 0.707107 − 0.707107i 0 −0.964841 + 2.01720i 0 −2.52090 0 − 1.00000i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 697.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f 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.cy.b yes 44
5.c odd 4 1 1560.2.bh.b ✓ 44
13.d odd 4 1 1560.2.bh.b ✓ 44
65.f even 4 1 inner 1560.2.cy.b yes 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.bh.b ✓ 44 5.c odd 4 1
1560.2.bh.b ✓ 44 13.d odd 4 1
1560.2.cy.b yes 44 1.a even 1 1 trivial
1560.2.cy.b yes 44 65.f even 4 1 inner