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Properties

Label 560.5.f.d
Level $560$
Weight $5$
Character orbit 560.f
Analytic conductor $57.887$
Analytic rank $0$
Dimension $32$
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] = [560,5,Mod(321,560)]
 
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, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.321");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 560.f (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: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{7} - 832 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{7} - 832 q^{9} - 144 q^{11} - 168 q^{21} - 4000 q^{25} + 1200 q^{29} + 600 q^{35} + 1632 q^{37} - 192 q^{39} + 6944 q^{43} - 7832 q^{49} - 6816 q^{51} + 2400 q^{53} + 3744 q^{57} - 2672 q^{63} + 3600 q^{65} - 18336 q^{67} + 21168 q^{71} - 21360 q^{77} + 1616 q^{79} - 16464 q^{81} + 6608 q^{91} + 45536 q^{93} - 3600 q^{95} + 120176 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1 0 − 15.9459i 0 − 11.1803i 0 11.4659 + 47.6396i 0 −173.272 0
321.2 0 − 15.8601i 0 11.1803i 0 −12.0083 − 47.5058i 0 −170.543 0
321.3 0 − 14.7187i 0 − 11.1803i 0 40.3727 − 27.7677i 0 −135.639 0
321.4 0 − 13.7259i 0 11.1803i 0 −40.4514 + 27.6530i 0 −107.399 0
321.5 0 − 13.2114i 0 11.1803i 0 48.0554 − 9.57486i 0 −93.5410 0
321.6 0 − 13.1729i 0 − 11.1803i 0 −32.1920 + 36.9415i 0 −92.5254 0
321.7 0 − 9.80434i 0 11.1803i 0 −14.3535 + 46.8506i 0 −15.1250 0
321.8 0 − 9.60387i 0 − 11.1803i 0 −9.22368 − 48.1240i 0 −11.2344 0
321.9 0 − 8.55676i 0 − 11.1803i 0 34.1791 − 35.1111i 0 7.78194 0
321.10 0 − 8.07634i 0 11.1803i 0 35.3670 + 33.9142i 0 15.7727 0
321.11 0 − 6.53615i 0 11.1803i 0 −13.0713 − 47.2244i 0 38.2787 0
321.12 0 − 6.05658i 0 11.1803i 0 −42.6426 − 24.1372i 0 44.3179 0
321.13 0 − 5.97820i 0 − 11.1803i 0 −45.1510 + 19.0365i 0 45.2611 0
321.14 0 − 2.76804i 0 − 11.1803i 0 −22.8323 − 43.3554i 0 73.3380 0
321.15 0 − 1.63668i 0 − 11.1803i 0 42.6409 + 24.1403i 0 78.3213 0
321.16 0 − 0.889751i 0 − 11.1803i 0 35.8450 + 33.4087i 0 80.2083 0
321.17 0 0.889751i 0 11.1803i 0 35.8450 − 33.4087i 0 80.2083 0
321.18 0 1.63668i 0 11.1803i 0 42.6409 − 24.1403i 0 78.3213 0
321.19 0 2.76804i 0 11.1803i 0 −22.8323 + 43.3554i 0 73.3380 0
321.20 0 5.97820i 0 11.1803i 0 −45.1510 − 19.0365i 0 45.2611 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b 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.f.d 32
4.b odd 2 1 280.5.f.a ✓ 32
7.b odd 2 1 inner 560.5.f.d 32
28.d even 2 1 280.5.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.5.f.a ✓ 32 4.b odd 2 1
280.5.f.a ✓ 32 28.d even 2 1
560.5.f.d 32 1.a even 1 1 trivial
560.5.f.d 32 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 1712 T_{3}^{30} + 1314068 T_{3}^{28} + 597848760 T_{3}^{26} + 179629788022 T_{3}^{24} + \cdots + 30\!\cdots\!76 \) acting on \(S_{5}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display