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Properties

Label 560.2.bl.a
Level $560$
Weight $2$
Character orbit 560.bl
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(267,560)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.267");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bl (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
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 8 q^{4} + 144 q^{9} - 16 q^{12} + 8 q^{16} + 28 q^{18} + 16 q^{19} - 36 q^{22} + 32 q^{26} - 36 q^{30} - 40 q^{32} - 32 q^{34} - 32 q^{36} - 56 q^{38} - 48 q^{40} + 20 q^{42} + 56 q^{44} - 32 q^{46} - 24 q^{48} + 8 q^{50} - 16 q^{51} - 16 q^{52} - 32 q^{54} - 24 q^{56} - 52 q^{58} - 32 q^{60} - 32 q^{61} + 40 q^{62} - 32 q^{64} + 32 q^{66} + 80 q^{68} + 32 q^{69} + 40 q^{72} + 16 q^{73} + 32 q^{74} - 24 q^{75} - 28 q^{78} + 72 q^{80} + 144 q^{81} + 24 q^{82} - 80 q^{83} + 32 q^{86} - 112 q^{87} + 40 q^{88} + 32 q^{90} - 16 q^{92} + 8 q^{94} - 80 q^{95} + 32 q^{96} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
267.1 −1.41384 + 0.0326401i −2.48131 1.99787 − 0.0922957i 1.99978 − 1.00045i 3.50817 − 0.0809903i −0.707107 − 0.707107i −2.82165 + 0.195702i 3.15689 −2.79470 + 1.47974i
267.2 −1.41376 + 0.0359421i −1.63763 1.99742 − 0.101627i −1.97240 − 1.05340i 2.31520 − 0.0588598i −0.707107 − 0.707107i −2.82021 + 0.215467i −0.318182 2.82635 + 1.41836i
267.3 −1.41369 + 0.0384129i 2.59192 1.99705 − 0.108608i 0.587579 − 2.15749i −3.66418 + 0.0995632i −0.707107 − 0.707107i −2.81904 + 0.230251i 3.71805 −0.747781 + 3.07259i
267.4 −1.40650 + 0.147475i 2.37954 1.95650 − 0.414848i 1.91514 + 1.15422i −3.34683 + 0.350923i 0.707107 + 0.707107i −2.69065 + 0.872021i 2.66220 −2.86387 − 1.34098i
267.5 −1.39954 − 0.203214i −0.534005 1.91741 + 0.568811i −0.00771645 − 2.23605i 0.747360 + 0.108517i 0.707107 + 0.707107i −2.56789 − 1.18572i −2.71484 −0.443598 + 3.13101i
267.6 −1.39085 − 0.256013i 0.334868 1.86892 + 0.712149i −0.401723 + 2.19969i −0.465750 − 0.0857303i −0.707107 − 0.707107i −2.41706 − 1.46896i −2.88786 1.12188 − 2.95658i
267.7 −1.36534 + 0.368567i −0.186664 1.72832 − 1.00644i 0.764144 + 2.10145i 0.254861 − 0.0687982i −0.707107 − 0.707107i −1.98880 + 2.01113i −2.96516 −1.81784 − 2.58756i
267.8 −1.36468 + 0.370996i 0.546698 1.72472 − 1.01259i −2.23178 + 0.138397i −0.746070 + 0.202823i 0.707107 + 0.707107i −1.97804 + 2.02173i −2.70112 2.99433 − 1.01685i
267.9 −1.32653 − 0.490211i 0.653961 1.51939 + 1.30056i 2.19612 + 0.420757i −0.867502 − 0.320579i 0.707107 + 0.707107i −1.37797 − 2.47006i −2.57233 −2.70697 − 1.63471i
267.10 −1.26598 + 0.630317i −2.31831 1.20540 − 1.59594i 0.465040 + 2.18718i 2.93493 − 1.46127i 0.707107 + 0.707107i −0.520065 + 2.78020i 2.37455 −1.96734 − 2.47579i
267.11 −1.24911 − 0.663122i −2.63429 1.12054 + 1.65662i −2.21909 − 0.275059i 3.29051 + 1.74686i 0.707107 + 0.707107i −0.301134 − 2.81235i 3.93949 2.58948 + 1.81510i
267.12 −1.22698 + 0.703226i −3.31270 1.01095 − 1.72568i −0.463660 − 2.18747i 4.06460 − 2.32957i 0.707107 + 0.707107i −0.0268617 + 2.82830i 7.97395 2.10719 + 2.35792i
267.13 −1.13370 − 0.845415i 1.65488 0.570547 + 1.91689i −2.16945 − 0.541746i −1.87614 − 1.39906i 0.707107 + 0.707107i 0.973740 − 2.65553i −0.261373 2.00150 + 2.44826i
267.14 −1.12140 − 0.861665i 1.39857 0.515067 + 1.93254i −1.32443 − 1.80163i −1.56835 − 1.20510i −0.707107 − 0.707107i 1.08761 − 2.61096i −1.04400 −0.0671870 + 3.16156i
267.15 −1.07546 − 0.918365i −1.03076 0.313212 + 1.97532i −1.74922 + 1.39292i 1.10853 + 0.946610i −0.707107 − 0.707107i 1.47722 − 2.41202i −1.93754 3.16042 + 0.108396i
267.16 −1.05597 + 0.940711i −0.615810 0.230127 − 1.98672i 2.16851 − 0.545484i 0.650274 − 0.579299i −0.707107 − 0.707107i 1.62592 + 2.31439i −2.62078 −1.77673 + 2.61596i
267.17 −1.04361 + 0.954397i 2.51711 0.178253 − 1.99204i 2.11634 + 0.721864i −2.62689 + 2.40232i −0.707107 − 0.707107i 1.71517 + 2.24904i 3.33584 −2.89759 + 1.26649i
267.18 −0.990624 + 1.00929i 3.44219 −0.0373263 − 1.99965i −1.65722 − 1.50121i −3.40992 + 3.47416i 0.707107 + 0.707107i 2.05520 + 1.94323i 8.84868 3.15683 − 0.185484i
267.19 −0.933601 + 1.06226i 1.37087 −0.256778 − 1.98345i −0.504798 + 2.17834i −1.27985 + 1.45622i 0.707107 + 0.707107i 2.34666 + 1.57898i −1.12071 −1.84268 − 2.56993i
267.20 −0.920909 − 1.07328i −1.27333 −0.303853 + 1.97678i 1.14059 − 1.92329i 1.17262 + 1.36664i 0.707107 + 0.707107i 2.40146 − 1.49432i −1.37862 −3.11461 + 0.547008i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 267.72
Significant digits:
Format:

Inner twists

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

Hecke kernels

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