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Properties

Label 560.2.bd.b
Level $560$
Weight $2$
Character orbit 560.bd
Analytic conductor $4.472$
Analytic rank $0$
Dimension $52$
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] = [560,2,Mod(141,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.141");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bd (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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 8 q^{4} + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 8 q^{4} + 12 q^{6} - 4 q^{10} - 4 q^{11} + 40 q^{12} + 8 q^{15} - 4 q^{16} + 8 q^{19} + 32 q^{22} + 12 q^{24} - 12 q^{26} - 24 q^{27} + 4 q^{28} + 4 q^{29} - 16 q^{34} + 4 q^{36} - 12 q^{37} + 16 q^{38} - 4 q^{42} + 36 q^{43} + 28 q^{44} - 16 q^{46} + 32 q^{48} - 52 q^{49} - 8 q^{51} - 80 q^{52} - 4 q^{53} + 48 q^{54} - 12 q^{56} - 56 q^{58} + 24 q^{59} - 16 q^{61} - 72 q^{62} - 68 q^{63} + 4 q^{64} + 40 q^{65} - 12 q^{66} - 12 q^{67} - 72 q^{69} - 8 q^{72} + 68 q^{74} - 4 q^{77} - 4 q^{78} - 16 q^{79} - 32 q^{80} - 116 q^{81} + 104 q^{82} + 16 q^{85} - 68 q^{86} - 48 q^{88} + 36 q^{90} + 64 q^{92} + 8 q^{93} - 72 q^{94} - 32 q^{95} - 52 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1 −1.36236 + 0.379431i 0.0811559 + 0.0811559i 1.71206 − 1.03385i 0.707107 − 0.707107i −0.141357 − 0.0797707i 1.00000i −1.94018 + 2.05808i − 2.98683i −0.695038 + 1.23163i
141.2 −1.32495 + 0.494478i −1.01907 − 1.01907i 1.51098 − 1.31032i −0.707107 + 0.707107i 1.85412 + 0.846307i 1.00000i −1.35405 + 2.48325i − 0.922998i 0.587232 − 1.28653i
141.3 −1.30578 + 0.543081i 2.37790 + 2.37790i 1.41013 − 1.41829i −0.707107 + 0.707107i −4.39640 − 1.81362i 1.00000i −1.07107 + 2.61779i 8.30879i 0.539310 − 1.30734i
141.4 −1.30541 − 0.543961i −2.02124 − 2.02124i 1.40821 + 1.42019i 0.707107 − 0.707107i 1.53908 + 3.73803i 1.00000i −1.06577 − 2.61995i 5.17082i −1.30771 + 0.538429i
141.5 −1.30240 − 0.551150i −1.24266 − 1.24266i 1.39247 + 1.43563i −0.707107 + 0.707107i 0.933546 + 2.30334i 1.00000i −1.02229 − 2.63722i 0.0884307i 1.31065 − 0.531211i
141.6 −0.961413 − 1.03715i 2.07558 + 2.07558i −0.151369 + 1.99426i 0.707107 − 0.707107i 0.157202 − 4.14819i 1.00000i 2.21388 − 1.76032i 5.61608i −1.41320 − 0.0535555i
141.7 −0.908304 + 1.08397i 0.523494 + 0.523494i −0.349967 − 1.96914i 0.707107 − 0.707107i −1.04294 + 0.0919583i 1.00000i 2.45236 + 1.40923i − 2.45191i 0.124212 + 1.40875i
141.8 −0.807148 + 1.16125i 2.00539 + 2.00539i −0.697026 − 1.87461i 0.707107 − 0.707107i −3.94742 + 0.710123i 1.00000i 2.73950 + 0.703661i 5.04319i 0.250392 + 1.39187i
141.9 −0.557588 + 1.29965i −2.40733 − 2.40733i −1.37819 − 1.44934i −0.707107 + 0.707107i 4.47099 − 1.78639i 1.00000i 2.65210 − 0.983032i 8.59045i −0.524718 − 1.31327i
141.10 −0.513830 − 1.31757i 0.363076 + 0.363076i −1.47196 + 1.35401i −0.707107 + 0.707107i 0.291817 − 0.664935i 1.00000i 2.54033 + 1.24367i − 2.73635i 1.29499 + 0.568327i
141.11 −0.488835 − 1.32704i −1.83016 − 1.83016i −1.52208 + 1.29741i 0.707107 − 0.707107i −1.53405 + 3.32334i 1.00000i 2.46576 + 1.38565i 3.69896i −1.28402 − 0.592702i
141.12 −0.171805 + 1.40374i −0.303944 − 0.303944i −1.94097 − 0.482338i −0.707107 + 0.707107i 0.478877 − 0.374439i 1.00000i 1.01054 − 2.64174i − 2.81524i −0.871109 − 1.11408i
141.13 −0.0789169 + 1.41201i −0.991372 − 0.991372i −1.98754 − 0.222863i 0.707107 − 0.707107i 1.47806 − 1.32159i 1.00000i 0.471536 − 2.78884i − 1.03436i 0.942639 + 1.05424i
141.14 −0.0328336 − 1.41383i 0.359066 + 0.359066i −1.99784 + 0.0928423i 0.707107 − 0.707107i 0.495870 − 0.519449i 1.00000i 0.196860 + 2.82157i − 2.74214i −1.02295 − 0.976514i
141.15 0.252525 − 1.39149i 1.86516 + 1.86516i −1.87246 − 0.702770i −0.707107 + 0.707107i 3.06634 − 2.12434i 1.00000i −1.45074 + 2.42804i 3.95763i 0.805366 + 1.16249i
141.16 0.328623 − 1.37550i −1.80727 − 1.80727i −1.78401 − 0.904043i −0.707107 + 0.707107i −3.07981 + 1.89199i 1.00000i −1.82978 + 2.15683i 3.53242i 0.740256 + 1.20500i
141.17 0.642045 + 1.26007i 0.975138 + 0.975138i −1.17556 + 1.61804i −0.707107 + 0.707107i −0.602660 + 1.85482i 1.00000i −2.79361 − 0.442425i − 1.09821i −1.34500 − 0.437010i
141.18 0.668656 − 1.24615i −1.21993 − 1.21993i −1.10580 − 1.66650i 0.707107 − 0.707107i −2.33593 + 0.704506i 1.00000i −2.81611 + 0.263684i − 0.0235503i −0.408353 − 1.35397i
141.19 0.865210 + 1.11867i −1.52217 − 1.52217i −0.502824 + 1.93576i 0.707107 − 0.707107i 0.385801 − 3.01979i 1.00000i −2.60052 + 1.11235i 1.63398i 1.40281 + 0.179220i
141.20 0.912154 + 1.08073i −1.81546 − 1.81546i −0.335949 + 1.97158i −0.707107 + 0.707107i 0.306040 − 3.61800i 1.00000i −2.43718 + 1.43532i 3.59179i −1.40918 − 0.119200i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bd.b ✓ 52
4.b odd 2 1 2240.2.bd.b 52
16.e even 4 1 inner 560.2.bd.b ✓ 52
16.f odd 4 1 2240.2.bd.b 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bd.b ✓ 52 1.a even 1 1 trivial
560.2.bd.b ✓ 52 16.e even 4 1 inner
2240.2.bd.b 52 4.b odd 2 1
2240.2.bd.b 52 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{52} + 8 T_{3}^{49} + 416 T_{3}^{48} + 72 T_{3}^{47} + 32 T_{3}^{46} + 2792 T_{3}^{45} + \cdots + 12845056 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display