Properties

Label 1560.2.dr.a
Level $1560$
Weight $2$
Character orbit 1560.dr
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(49,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.49");
 
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.dr (of order \(6\), 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 \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{5} + 2 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{5} + 2 q^{7} + 22 q^{9} - 6 q^{11} - 4 q^{13} + 18 q^{17} - 6 q^{19} - 6 q^{23} + 2 q^{25} - 2 q^{29} - 6 q^{33} - 16 q^{37} - 2 q^{39} - 6 q^{41} - 36 q^{43} - 4 q^{45} - 48 q^{47} - 36 q^{49} + 24 q^{51} - 24 q^{55} + 24 q^{57} - 14 q^{61} - 2 q^{63} + 26 q^{65} - 4 q^{67} - 8 q^{69} + 44 q^{73} - 16 q^{75} + 28 q^{79} - 22 q^{81} - 56 q^{83} - 44 q^{85} - 6 q^{87} + 18 q^{89} + 30 q^{91} + 8 q^{93} - 34 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −0.866025 + 0.500000i 0 −2.22388 + 0.233190i 0 2.24062 3.88087i 0 0.500000 0.866025i 0
49.2 0 −0.866025 + 0.500000i 0 −1.70064 1.45184i 0 1.91976 3.32513i 0 0.500000 0.866025i 0
49.3 0 −0.866025 + 0.500000i 0 −2.16416 0.562520i 0 −1.46016 + 2.52908i 0 0.500000 0.866025i 0
49.4 0 −0.866025 + 0.500000i 0 0.455587 + 2.18916i 0 1.28750 2.23001i 0 0.500000 0.866025i 0
49.5 0 −0.866025 + 0.500000i 0 1.63725 1.52296i 0 1.13279 1.96206i 0 0.500000 0.866025i 0
49.6 0 −0.866025 + 0.500000i 0 −1.75606 + 1.38429i 0 −1.15793 + 2.00559i 0 0.500000 0.866025i 0
49.7 0 −0.866025 + 0.500000i 0 2.23438 0.0868459i 0 1.01600 1.75977i 0 0.500000 0.866025i 0
49.8 0 −0.866025 + 0.500000i 0 −0.0598114 + 2.23527i 0 −0.402748 + 0.697580i 0 0.500000 0.866025i 0
49.9 0 −0.866025 + 0.500000i 0 1.24679 1.85621i 0 0.168505 0.291860i 0 0.500000 0.866025i 0
49.10 0 −0.866025 + 0.500000i 0 −0.260086 2.22089i 0 −1.86493 + 3.23015i 0 0.500000 0.866025i 0
49.11 0 −0.866025 + 0.500000i 0 2.09061 + 0.793323i 0 −2.37941 + 4.12126i 0 0.500000 0.866025i 0
49.12 0 0.866025 0.500000i 0 2.20743 0.356749i 0 2.27952 3.94824i 0 0.500000 0.866025i 0
49.13 0 0.866025 0.500000i 0 −1.08602 + 1.95462i 0 1.85933 3.22046i 0 0.500000 0.866025i 0
49.14 0 0.866025 0.500000i 0 −1.82229 + 1.29586i 0 −1.66961 + 2.89185i 0 0.500000 0.866025i 0
49.15 0 0.866025 0.500000i 0 −1.49747 1.66059i 0 1.56480 2.71031i 0 0.500000 0.866025i 0
49.16 0 0.866025 0.500000i 0 0.987261 2.00632i 0 −0.931563 + 1.61351i 0 0.500000 0.866025i 0
49.17 0 0.866025 0.500000i 0 0.644860 2.14106i 0 0.855874 1.48242i 0 0.500000 0.866025i 0
49.18 0 0.866025 0.500000i 0 1.50795 + 1.65109i 0 −0.673071 + 1.16579i 0 0.500000 0.866025i 0
49.19 0 0.866025 0.500000i 0 −0.868720 + 2.06042i 0 −0.489404 + 0.847672i 0 0.500000 0.866025i 0
49.20 0 0.866025 0.500000i 0 −2.23603 0.0122650i 0 0.349261 0.604938i 0 0.500000 0.866025i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.dr.a 44
5.b even 2 1 1560.2.dr.b yes 44
13.e even 6 1 1560.2.dr.b yes 44
65.l even 6 1 inner 1560.2.dr.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.dr.a 44 1.a even 1 1 trivial
1560.2.dr.a 44 65.l even 6 1 inner
1560.2.dr.b yes 44 5.b even 2 1
1560.2.dr.b yes 44 13.e even 6 1