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Properties

Label 280.2.bf.a
Level $280$
Weight $2$
Character orbit 280.bf
Analytic conductor $2.236$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,2,Mod(109,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.bf (of order \(6\), 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: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{4} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{4} - 40 q^{9} + 6 q^{10} - 6 q^{14} + 4 q^{15} + 2 q^{16} - 24 q^{20} - 8 q^{24} - 2 q^{25} - 14 q^{26} + 14 q^{30} - 28 q^{31} + 24 q^{34} + 60 q^{36} - 16 q^{39} - 32 q^{40} - 16 q^{41} - 2 q^{44} - 42 q^{46} - 8 q^{49} - 12 q^{50} + 8 q^{54} - 28 q^{55} + 36 q^{56} + 10 q^{60} + 52 q^{64} - 12 q^{65} - 44 q^{66} - 48 q^{70} + 16 q^{71} - 42 q^{74} - 36 q^{76} - 4 q^{79} - 40 q^{80} - 20 q^{81} + 12 q^{84} - 24 q^{86} + 4 q^{89} - 96 q^{90} - 22 q^{94} - 34 q^{95} + 80 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41234 − 0.0727260i −1.37008 + 2.37305i 1.98942 + 0.205428i −2.18145 + 0.491206i 2.10761 − 3.25192i −2.02537 − 1.70232i −2.79480 − 0.434817i −2.25425 − 3.90447i 3.11668 − 0.535104i
109.2 −1.39743 + 0.217250i 0.284947 − 0.493542i 1.90560 − 0.607183i 0.172645 + 2.22939i −0.290970 + 0.751594i −2.45778 + 0.979438i −2.53103 + 1.26249i 1.33761 + 2.31681i −0.725596 − 3.07791i
109.3 −1.38464 − 0.287716i 1.62181 − 2.80906i 1.83444 + 0.796764i −0.194592 + 2.22758i −3.05383 + 3.42291i 1.16173 − 2.37705i −2.31079 − 1.63103i −3.76055 − 6.51347i 0.910350 − 3.02841i
109.4 −1.38305 − 0.295272i −0.496594 + 0.860127i 1.82563 + 0.816749i 1.99653 − 1.00690i 0.940784 − 1.04296i 2.24690 − 1.39695i −2.28377 − 1.66866i 1.00679 + 1.74381i −3.05861 + 0.803068i
109.5 −1.34506 + 0.436831i 0.866705 − 1.50118i 1.61836 − 1.17513i 1.99503 − 1.00988i −0.510007 + 2.39777i 1.01592 + 2.44293i −1.66345 + 2.28756i −0.00235378 − 0.00407687i −2.24228 + 2.22984i
109.6 −1.32820 − 0.485693i 0.390744 − 0.676788i 1.52820 + 1.29019i −2.21752 − 0.287435i −0.847695 + 0.709125i 1.02493 + 2.43916i −1.40312 − 2.45586i 1.19464 + 2.06917i 2.80569 + 1.45880i
109.7 −1.29160 + 0.576005i 0.755963 − 1.30937i 1.33644 − 1.48793i −1.54124 − 1.62006i −0.222196 + 2.12661i 0.727432 − 2.54379i −0.869078 + 2.69160i 0.357040 + 0.618411i 2.92382 + 1.20470i
109.8 −1.19702 − 0.753089i −1.40092 + 2.42646i 0.865713 + 1.80293i 0.791109 + 2.09145i 3.50426 − 1.84950i 2.13463 + 1.56312i 0.321488 − 2.81010i −2.42513 − 4.20044i 0.628073 − 3.09928i
109.9 −1.14463 + 0.830552i −0.755963 + 1.30937i 0.620368 − 1.90135i 1.54124 + 1.62006i −0.222196 − 2.12661i 0.727432 − 2.54379i 0.869078 + 2.69160i 0.357040 + 0.618411i −3.10969 − 0.574295i
109.10 −1.05084 + 0.946438i −0.866705 + 1.50118i 0.208510 − 1.98910i −1.99503 + 1.00988i −0.510007 − 2.39777i 1.01592 + 2.44293i 1.66345 + 2.28756i −0.00235378 − 0.00407687i 1.14066 − 2.94939i
109.11 −1.02388 − 0.975536i 0.299892 − 0.519428i 0.0966599 + 1.99766i 2.04027 + 0.915040i −0.813774 + 0.239277i −2.46748 − 0.954749i 1.84982 − 2.13966i 1.32013 + 2.28653i −1.19634 − 2.92725i
109.12 −0.912095 − 1.08078i −0.945048 + 1.63687i −0.336164 + 1.97155i −0.0512873 − 2.23548i 2.63107 − 0.471595i −1.18371 + 2.36619i 2.43742 − 1.43492i −0.286230 − 0.495764i −2.36928 + 2.09440i
109.13 −0.886858 + 1.10158i −0.284947 + 0.493542i −0.426967 − 1.95389i −0.172645 − 2.22939i −0.290970 − 0.751594i −2.45778 + 0.979438i 2.53103 + 1.26249i 1.33761 + 2.31681i 2.60897 + 1.78697i
109.14 −0.805587 − 1.16234i 1.22223 − 2.11697i −0.702060 + 1.87273i 1.10428 − 1.94437i −3.44524 + 0.284754i 2.60788 − 0.446057i 2.74231 − 0.692614i −1.48770 − 2.57677i −3.14961 + 0.282810i
109.15 −0.643189 + 1.25949i 1.37008 − 2.37305i −1.17262 − 1.62018i 2.18145 − 0.491206i 2.10761 + 3.25192i −2.02537 − 1.70232i 2.79480 − 0.434817i −2.25425 − 3.90447i −0.784414 + 3.06344i
109.16 −0.603821 − 1.27883i 1.22223 − 2.11697i −1.27080 + 1.54437i −2.23601 − 0.0158492i −3.44524 − 0.284754i −2.60788 + 0.446057i 2.74231 + 0.692614i −1.48770 − 2.57677i 1.32988 + 2.86904i
109.17 −0.479934 − 1.33029i −0.945048 + 1.63687i −1.53933 + 1.27690i −1.91034 − 1.16216i 2.63107 + 0.471595i 1.18371 − 2.36619i 2.43742 + 1.43492i −0.286230 − 0.495764i −0.629165 + 3.09906i
109.18 −0.443149 + 1.34299i −1.62181 + 2.80906i −1.60724 − 1.19029i 0.194592 − 2.22758i −3.05383 − 3.42291i 1.16173 − 2.37705i 2.31079 − 1.63103i −3.76055 − 6.51347i 2.90539 + 1.24849i
109.19 −0.435810 + 1.34539i 0.496594 − 0.860127i −1.62014 − 1.17267i −1.99653 + 1.00690i 0.940784 + 1.04296i 2.24690 − 1.39695i 2.28377 − 1.66866i 1.00679 + 1.74381i −0.484562 − 3.12493i
109.20 −0.332899 − 1.37447i 0.299892 − 0.519428i −1.77836 + 0.915122i −0.227688 + 2.22445i −0.813774 − 0.239277i 2.46748 + 0.954749i 1.84982 + 2.13966i 1.32013 + 2.28653i 3.13324 − 0.427565i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
35.j even 6 1 inner
40.f even 2 1 inner
56.p even 6 1 inner
280.bf even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bf.a ✓ 88
4.b odd 2 1 1120.2.bv.a 88
5.b even 2 1 inner 280.2.bf.a ✓ 88
7.c even 3 1 inner 280.2.bf.a ✓ 88
8.b even 2 1 inner 280.2.bf.a ✓ 88
8.d odd 2 1 1120.2.bv.a 88
20.d odd 2 1 1120.2.bv.a 88
28.g odd 6 1 1120.2.bv.a 88
35.j even 6 1 inner 280.2.bf.a ✓ 88
40.e odd 2 1 1120.2.bv.a 88
40.f even 2 1 inner 280.2.bf.a ✓ 88
56.k odd 6 1 1120.2.bv.a 88
56.p even 6 1 inner 280.2.bf.a ✓ 88
140.p odd 6 1 1120.2.bv.a 88
280.bf even 6 1 inner 280.2.bf.a ✓ 88
280.bi odd 6 1 1120.2.bv.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bf.a ✓ 88 1.a even 1 1 trivial
280.2.bf.a ✓ 88 5.b even 2 1 inner
280.2.bf.a ✓ 88 7.c even 3 1 inner
280.2.bf.a ✓ 88 8.b even 2 1 inner
280.2.bf.a ✓ 88 35.j even 6 1 inner
280.2.bf.a ✓ 88 40.f even 2 1 inner
280.2.bf.a ✓ 88 56.p even 6 1 inner
280.2.bf.a ✓ 88 280.bf even 6 1 inner
1120.2.bv.a 88 4.b odd 2 1
1120.2.bv.a 88 8.d odd 2 1
1120.2.bv.a 88 20.d odd 2 1
1120.2.bv.a 88 28.g odd 6 1
1120.2.bv.a 88 40.e odd 2 1
1120.2.bv.a 88 56.k odd 6 1
1120.2.bv.a 88 140.p odd 6 1
1120.2.bv.a 88 280.bi odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(280, [\chi])\).