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Properties

Label 560.2.bd.a
Level $560$
Weight $2$
Character orbit 560.bd
Analytic conductor $4.472$
Analytic rank $0$
Dimension $44$
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: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 12 q^{4} + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 12 q^{4} + 12 q^{6} - 4 q^{10} + 12 q^{11} - 16 q^{12} + 4 q^{14} + 8 q^{15} - 8 q^{16} - 20 q^{18} + 8 q^{19} - 36 q^{22} + 12 q^{24} + 44 q^{26} - 24 q^{27} - 4 q^{28} + 12 q^{29} - 40 q^{32} - 16 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} + 4 q^{42} - 44 q^{43} - 32 q^{44} - 16 q^{46} - 32 q^{48} - 44 q^{49} + 4 q^{50} - 8 q^{51} + 16 q^{52} - 12 q^{53} - 80 q^{54} + 8 q^{56} + 4 q^{58} + 24 q^{59} - 16 q^{61} + 28 q^{63} + 72 q^{64} - 40 q^{65} - 20 q^{66} - 28 q^{67} + 56 q^{68} + 40 q^{69} + 12 q^{72} + 24 q^{74} - 12 q^{77} + 84 q^{78} - 16 q^{79} + 20 q^{81} + 48 q^{82} + 16 q^{85} - 64 q^{86} + 28 q^{88} - 36 q^{90} - 16 q^{92} + 88 q^{93} + 96 q^{94} - 32 q^{95} + 48 q^{96} + 28 q^{99}+O(q^{100}) \) 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 −1.41203 − 0.0784738i 0.257753 + 0.257753i 1.98768 + 0.221615i −0.707107 + 0.707107i −0.343730 − 0.384184i − 1.00000i −2.78929 − 0.468910i − 2.86713i 1.05395 − 0.942970i
141.2 −1.40016 − 0.198865i 1.92081 + 1.92081i 1.92091 + 0.556887i 0.707107 − 0.707107i −2.30746 − 3.07142i − 1.00000i −2.57883 − 1.16173i 4.37899i −1.13068 + 0.849445i
141.3 −1.39986 + 0.200956i −2.22143 − 2.22143i 1.91923 − 0.562622i −0.707107 + 0.707107i 3.55611 + 2.66329i − 1.00000i −2.57360 + 1.17328i 6.86952i 0.847755 − 1.13195i
141.4 −1.37680 + 0.323129i −1.67958 − 1.67958i 1.79118 − 0.889770i 0.707107 − 0.707107i 2.85517 + 1.76973i − 1.00000i −2.17859 + 1.80382i 2.64195i −0.745060 + 1.20203i
141.5 −1.16029 + 0.808533i 0.925827 + 0.925827i 0.692549 − 1.87627i −0.707107 + 0.707107i −1.82279 − 0.325667i − 1.00000i 0.713464 + 2.73696i − 1.28569i 0.248730 − 1.39217i
141.6 −1.04918 − 0.948274i 0.448521 + 0.448521i 0.201553 + 1.98982i 0.707107 − 0.707107i −0.0452579 − 0.895900i − 1.00000i 1.67543 − 2.27880i − 2.59766i −1.41241 + 0.0713505i
141.7 −0.914710 + 1.07857i −1.42818 − 1.42818i −0.326610 − 1.97315i 0.707107 − 0.707107i 2.84676 − 0.234016i − 1.00000i 2.42693 + 1.45259i 1.07940i 0.115864 + 1.40946i
141.8 −0.630992 + 1.26564i 0.659301 + 0.659301i −1.20370 − 1.59722i 0.707107 − 0.707107i −1.25045 + 0.418425i − 1.00000i 2.78103 − 0.515617i − 2.13064i 0.448765 + 1.34112i
141.9 −0.590325 + 1.28511i −0.839605 − 0.839605i −1.30303 − 1.51727i −0.707107 + 0.707107i 1.57463 − 0.583348i − 1.00000i 2.71908 − 0.778863i − 1.59013i −0.491290 − 1.32614i
141.10 −0.560807 − 1.29827i 0.693100 + 0.693100i −1.37099 + 1.45615i −0.707107 + 0.707107i 0.511134 − 1.28852i − 1.00000i 2.65934 + 0.963293i − 2.03922i 1.31456 + 0.521463i
141.11 −0.551277 − 1.30234i −1.16279 − 1.16279i −1.39219 + 1.43590i 0.707107 − 0.707107i −0.873327 + 2.15536i − 1.00000i 2.63752 + 1.02152i − 0.295857i −1.31071 − 0.531083i
141.12 0.381101 + 1.36190i −0.605289 − 0.605289i −1.70952 + 1.03804i −0.707107 + 0.707107i 0.593665 − 1.05502i − 1.00000i −2.06521 − 1.93260i − 2.26725i −1.23249 − 0.693527i
141.13 0.504607 + 1.32113i 2.22016 + 2.22016i −1.49074 + 1.33330i 0.707107 − 0.707107i −1.81280 + 4.05342i − 1.00000i −2.51369 − 1.29667i 6.85821i 1.29099 + 0.577366i
141.14 0.517296 − 1.31621i −0.296675 − 0.296675i −1.46481 − 1.36174i −0.707107 + 0.707107i −0.543956 + 0.237018i − 1.00000i −2.55007 + 1.22357i − 2.82397i 0.564917 + 1.29648i
141.15 0.817344 − 1.15410i 1.28029 + 1.28029i −0.663897 − 1.88660i 0.707107 − 0.707107i 2.52403 − 0.431148i − 1.00000i −2.71995 − 0.775794i 0.278310i −0.238123 − 1.39402i
141.16 1.13939 + 0.837723i −1.78962 − 1.78962i 0.596439 + 1.90899i −0.707107 + 0.707107i −0.539875 − 3.53828i − 1.00000i −0.919631 + 2.67475i 3.40546i −1.39803 + 0.213314i
141.17 1.17062 − 0.793500i −1.25769 − 1.25769i 0.740715 − 1.85778i 0.707107 − 0.707107i −2.47025 − 0.474301i − 1.00000i −0.607051 − 2.76252i 0.163545i 0.266666 − 1.38884i
141.18 1.18248 + 0.775723i 1.63220 + 1.63220i 0.796507 + 1.83455i −0.707107 + 0.707107i 0.663904 + 3.19617i − 1.00000i −0.481253 + 2.78718i 2.32815i −1.38466 + 0.287619i
141.19 1.20137 − 0.746130i 2.18833 + 2.18833i 0.886580 − 1.79276i −0.707107 + 0.707107i 4.26177 + 0.996215i − 1.00000i −0.272519 − 2.81527i 6.57756i −0.321903 + 1.37709i
141.20 1.32415 + 0.496608i −0.576404 − 0.576404i 1.50676 + 1.31517i 0.707107 − 0.707107i −0.477000 − 1.04949i − 1.00000i 1.34206 + 2.48976i − 2.33552i 1.28747 − 0.585162i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.22
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.a ✓ 44
4.b odd 2 1 2240.2.bd.a 44
16.e even 4 1 inner 560.2.bd.a ✓ 44
16.f odd 4 1 2240.2.bd.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bd.a ✓ 44 1.a even 1 1 trivial
560.2.bd.a ✓ 44 16.e even 4 1 inner
2240.2.bd.a 44 4.b odd 2 1
2240.2.bd.a 44 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{44} + 8 T_{3}^{41} + 256 T_{3}^{40} + 72 T_{3}^{39} + 32 T_{3}^{38} + 1480 T_{3}^{37} + \cdots + 10137856 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display