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Properties

Label 560.5.p.j
Level $560$
Weight $5$
Character orbit 560.p
Analytic conductor $57.887$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,5,Mod(209,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.209");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 560.p (of order \(2\), degree \(1\), not 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: \(57.8871793270\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 1328 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 1328 q^{9} + 144 q^{11} + 352 q^{15} - 808 q^{21} - 560 q^{25} + 1584 q^{29} + 1464 q^{35} - 4736 q^{39} - 4440 q^{49} + 4128 q^{51} - 2928 q^{65} - 816 q^{71} + 5776 q^{79} + 26496 q^{81} + 4896 q^{85} + 11088 q^{91} - 27120 q^{95} + 30576 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −16.3049 0 −19.7058 + 15.3844i 0 −30.3215 + 38.4916i 0 184.850 0
209.2 0 −16.3049 0 −19.7058 − 15.3844i 0 −30.3215 − 38.4916i 0 184.850 0
209.3 0 −16.1918 0 9.61199 − 23.0783i 0 31.4357 + 37.5872i 0 181.173 0
209.4 0 −16.1918 0 9.61199 + 23.0783i 0 31.4357 − 37.5872i 0 181.173 0
209.5 0 −15.0659 0 24.8187 + 3.00531i 0 −11.3989 + 47.6557i 0 145.980 0
209.6 0 −15.0659 0 24.8187 − 3.00531i 0 −11.3989 − 47.6557i 0 145.980 0
209.7 0 −11.1000 0 8.73979 + 23.4226i 0 43.4428 + 22.6654i 0 42.2108 0
209.8 0 −11.1000 0 8.73979 − 23.4226i 0 43.4428 − 22.6654i 0 42.2108 0
209.9 0 −10.8422 0 −22.1336 − 11.6234i 0 45.7416 − 17.5699i 0 36.5525 0
209.10 0 −10.8422 0 −22.1336 + 11.6234i 0 45.7416 + 17.5699i 0 36.5525 0
209.11 0 −10.7866 0 −19.2084 − 16.0012i 0 −38.9244 − 29.7639i 0 35.3508 0
209.12 0 −10.7866 0 −19.2084 + 16.0012i 0 −38.9244 + 29.7639i 0 35.3508 0
209.13 0 −8.38692 0 −15.4256 − 19.6736i 0 29.1949 + 39.3530i 0 −10.6596 0
209.14 0 −8.38692 0 −15.4256 + 19.6736i 0 29.1949 − 39.3530i 0 −10.6596 0
209.15 0 −7.95643 0 14.7379 + 20.1939i 0 −48.9359 + 2.50515i 0 −17.6953 0
209.16 0 −7.95643 0 14.7379 − 20.1939i 0 −48.9359 − 2.50515i 0 −17.6953 0
209.17 0 −6.92007 0 −2.98272 − 24.8214i 0 −21.3263 + 44.1156i 0 −33.1126 0
209.18 0 −6.92007 0 −2.98272 + 24.8214i 0 −21.3263 − 44.1156i 0 −33.1126 0
209.19 0 −2.34812 0 23.9739 + 7.08882i 0 48.3030 + 8.23527i 0 −75.4863 0
209.20 0 −2.34812 0 23.9739 − 7.08882i 0 48.3030 − 8.23527i 0 −75.4863 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.p.j 48
4.b odd 2 1 280.5.p.a ✓ 48
5.b even 2 1 inner 560.5.p.j 48
7.b odd 2 1 inner 560.5.p.j 48
20.d odd 2 1 280.5.p.a ✓ 48
28.d even 2 1 280.5.p.a ✓ 48
35.c odd 2 1 inner 560.5.p.j 48
140.c even 2 1 280.5.p.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.5.p.a ✓ 48 4.b odd 2 1
280.5.p.a ✓ 48 20.d odd 2 1
280.5.p.a ✓ 48 28.d even 2 1
280.5.p.a ✓ 48 140.c even 2 1
560.5.p.j 48 1.a even 1 1 trivial
560.5.p.j 48 5.b even 2 1 inner
560.5.p.j 48 7.b odd 2 1 inner
560.5.p.j 48 35.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 1304 T_{3}^{22} + 727826 T_{3}^{20} - 228066076 T_{3}^{18} + 44254082681 T_{3}^{16} + \cdots + 95\!\cdots\!44 \) acting on \(S_{5}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display