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Properties

Label 560.2.t.a
Level $560$
Weight $2$
Character orbit 560.t
Analytic conductor $4.472$
Analytic rank $0$
Dimension $144$
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(43,560)]
 
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([2, 1, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.t (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: \(144\)
Relative dimension: \(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 8 q^{4} - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 8 q^{4} - 144 q^{9} - 16 q^{12} + 8 q^{16} - 28 q^{18} - 16 q^{19} + 4 q^{22} + 32 q^{26} - 36 q^{30} + 32 q^{34} - 32 q^{36} - 56 q^{38} + 8 q^{40} - 20 q^{42} + 64 q^{43} - 56 q^{44} - 32 q^{46} - 80 q^{48} - 48 q^{50} - 16 q^{51} - 16 q^{52} + 32 q^{54} - 24 q^{56} + 52 q^{58} + 72 q^{60} - 32 q^{61} - 40 q^{62} + 32 q^{64} + 32 q^{66} - 24 q^{68} - 32 q^{69} + 80 q^{72} - 16 q^{73} - 32 q^{74} - 88 q^{75} + 76 q^{78} + 128 q^{80} + 144 q^{81} + 56 q^{82} + 32 q^{86} - 112 q^{87} + 88 q^{88} + 88 q^{90} - 136 q^{92} - 8 q^{94} + 80 q^{95} + 32 q^{96} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41270 − 0.0654069i 1.49919i 1.99144 + 0.184801i 1.45710 − 1.69613i 0.0980575 − 2.11791i −0.707107 + 0.707107i −2.80123 − 0.391322i 0.752424 −2.16939 + 2.30082i
43.2 −1.41154 − 0.0869980i − 1.58286i 1.98486 + 0.245601i 1.89869 − 1.18110i −0.137706 + 2.23427i 0.707107 − 0.707107i −2.78034 − 0.519354i 0.494538 −2.78282 + 1.50198i
43.3 −1.40839 + 0.128258i − 0.748637i 1.96710 − 0.361275i 0.813660 + 2.08278i 0.0960190 + 1.05437i −0.707107 + 0.707107i −2.72410 + 0.761111i 2.43954 −1.41308 − 2.82899i
43.4 −1.40199 − 0.185504i − 1.80332i 1.93118 + 0.520150i −0.550864 + 2.16715i −0.334523 + 2.52824i 0.707107 − 0.707107i −2.61101 − 1.08749i −0.251963 1.17432 − 2.93615i
43.5 −1.39923 + 0.205299i − 0.670195i 1.91570 − 0.574522i −1.47177 − 1.68342i 0.137590 + 0.937758i −0.707107 + 0.707107i −2.56257 + 1.19718i 2.55084 2.40495 + 2.05334i
43.6 −1.39015 − 0.259757i 2.75657i 1.86505 + 0.722204i −1.95957 + 1.07707i 0.716039 − 3.83206i −0.707107 + 0.707107i −2.40511 − 1.48843i −4.59869 3.00388 − 0.988279i
43.7 −1.37092 + 0.347225i 2.77381i 1.75887 − 0.952039i 1.05275 − 1.97274i −0.963136 − 3.80268i 0.707107 − 0.707107i −2.08071 + 1.91590i −4.69401 −0.758256 + 3.07002i
43.8 −1.36754 − 0.360316i 3.13969i 1.74034 + 0.985496i 1.64497 + 1.51462i 1.13128 − 4.29365i 0.707107 − 0.707107i −2.02490 − 1.97478i −6.85762 −1.70383 − 2.66401i
43.9 −1.36029 + 0.386778i 1.62010i 1.70080 − 1.05227i −2.19508 − 0.426188i −0.626621 − 2.20382i 0.707107 − 0.707107i −1.90660 + 2.08923i 0.375266 3.15079 − 0.269268i
43.10 −1.33923 + 0.454387i − 3.28908i 1.58707 − 1.21706i 2.05100 − 0.890731i 1.49451 + 4.40483i −0.707107 + 0.707107i −1.57243 + 2.35106i −7.81803 −2.34202 + 2.12484i
43.11 −1.29547 − 0.567247i 0.154841i 1.35646 + 1.46970i −0.497515 − 2.18002i 0.0878329 − 0.200591i 0.707107 − 0.707107i −0.923567 − 2.67339i 2.97602 −0.592095 + 3.10635i
43.12 −1.22105 − 0.713475i − 2.06291i 0.981908 + 1.74237i 1.83957 + 1.27121i −1.47183 + 2.51891i −0.707107 + 0.707107i 0.0441824 − 2.82808i −1.25559 −1.33923 − 2.86469i
43.13 −1.18735 + 0.768249i − 0.0697127i 0.819588 − 1.82436i 2.20646 + 0.362656i 0.0535567 + 0.0827732i 0.707107 − 0.707107i 0.428424 + 2.79579i 2.99514 −2.89845 + 1.26451i
43.14 −1.16689 − 0.798984i 1.17950i 0.723250 + 1.86465i 0.0146410 + 2.23602i 0.942403 − 1.37635i 0.707107 − 0.707107i 0.645873 − 2.75370i 1.60878 1.76946 − 2.62088i
43.15 −1.11562 + 0.869134i − 2.40982i 0.489211 − 1.93925i −2.07668 + 0.829085i 2.09446 + 2.68844i −0.707107 + 0.707107i 1.13969 + 2.58865i −2.80723 1.59620 − 2.72986i
43.16 −1.09978 − 0.889090i 2.37538i 0.419039 + 1.95561i −0.533053 − 2.17160i 2.11192 − 2.61239i −0.707107 + 0.707107i 1.27786 − 2.52331i −2.64241 −1.34451 + 2.86222i
43.17 −1.09013 + 0.900895i 2.66085i 0.376777 − 1.96419i 1.75709 + 1.38298i −2.39714 − 2.90067i −0.707107 + 0.707107i 1.35879 + 2.48066i −4.08010 −3.16138 + 0.0753290i
43.18 −1.07328 + 0.920909i − 1.27333i 0.303853 − 1.97678i −1.92329 + 1.14059i 1.17262 + 1.36664i 0.707107 − 0.707107i 1.49432 + 2.40146i 1.37862 1.01385 − 2.99535i
43.19 −0.975664 − 1.02376i − 0.864258i −0.0961601 + 1.99769i −2.07047 − 0.844473i −0.884791 + 0.843225i −0.707107 + 0.707107i 2.13897 − 1.85063i 2.25306 1.15555 + 2.94359i
43.20 −0.918365 + 1.07546i − 1.03076i −0.313212 − 1.97532i 1.39292 − 1.74922i 1.10853 + 0.946610i −0.707107 + 0.707107i 2.41202 + 1.47722i 1.93754 0.601999 + 3.10445i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j 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.t.a ✓ 144
5.c odd 4 1 560.2.bl.a yes 144
16.f odd 4 1 560.2.bl.a yes 144
80.j even 4 1 inner 560.2.t.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.t.a ✓ 144 1.a even 1 1 trivial
560.2.t.a ✓ 144 80.j even 4 1 inner
560.2.bl.a yes 144 5.c odd 4 1
560.2.bl.a yes 144 16.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(560, [\chi])\).