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Properties

Label 280.2.l.a
Level $280$
Weight $2$
Character orbit 280.l
Analytic conductor $2.236$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,2,Mod(29,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("280.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.l (of order \(2\), degree \(1\), 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: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{4} - 4 q^{6} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{4} - 4 q^{6} + 36 q^{9} - 8 q^{10} + 20 q^{16} - 24 q^{20} - 48 q^{24} + 4 q^{25} - 4 q^{26} + 4 q^{30} - 16 q^{31} + 12 q^{34} - 20 q^{36} - 32 q^{39} + 16 q^{40} - 8 q^{41} + 56 q^{44} - 36 q^{49} - 12 q^{50} - 52 q^{54} - 32 q^{55} + 12 q^{56} - 20 q^{60} - 20 q^{64} - 24 q^{65} - 28 q^{66} - 12 q^{70} + 56 q^{71} - 24 q^{74} + 48 q^{76} + 24 q^{79} + 64 q^{80} + 36 q^{81} + 24 q^{86} - 40 q^{89} - 52 q^{90} - 92 q^{94} + 40 q^{95} + 48 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.38397 − 0.290898i −1.83171 1.83076 + 0.805189i 1.41945 − 1.72776i 2.53504 + 0.532841i 1.00000i −2.29949 − 1.64692i 0.355171 −2.46708 + 1.97826i
29.2 −1.38397 + 0.290898i −1.83171 1.83076 − 0.805189i 1.41945 + 1.72776i 2.53504 − 0.532841i − 1.00000i −2.29949 + 1.64692i 0.355171 −2.46708 − 1.97826i
29.3 −1.37343 − 0.337187i 3.10393 1.77261 + 0.926204i −0.0152300 − 2.23602i −4.26302 − 1.04660i 1.00000i −2.12225 − 1.86978i 6.63436 −0.733038 + 3.07614i
29.4 −1.37343 + 0.337187i 3.10393 1.77261 − 0.926204i −0.0152300 + 2.23602i −4.26302 + 1.04660i − 1.00000i −2.12225 + 1.86978i 6.63436 −0.733038 − 3.07614i
29.5 −1.36669 − 0.363546i 0.460762 1.73567 + 0.993708i −2.19557 + 0.423658i −0.629718 − 0.167508i − 1.00000i −2.01086 − 1.98908i −2.78770 3.15467 + 0.219182i
29.6 −1.36669 + 0.363546i 0.460762 1.73567 − 0.993708i −2.19557 − 0.423658i −0.629718 + 0.167508i 1.00000i −2.01086 + 1.98908i −2.78770 3.15467 − 0.219182i
29.7 −1.23615 − 0.686976i −0.359051 1.05613 + 1.69841i 0.565870 + 2.16328i 0.443841 + 0.246660i 1.00000i −0.138765 − 2.82502i −2.87108 0.786624 − 3.06288i
29.8 −1.23615 + 0.686976i −0.359051 1.05613 − 1.69841i 0.565870 − 2.16328i 0.443841 − 0.246660i − 1.00000i −0.138765 + 2.82502i −2.87108 0.786624 + 3.06288i
29.9 −1.12571 − 0.856025i 1.65329 0.534444 + 1.92727i 2.21195 + 0.327549i −1.86112 − 1.41526i − 1.00000i 1.04816 − 2.62704i −0.266637 −2.20962 − 2.26221i
29.10 −1.12571 + 0.856025i 1.65329 0.534444 − 1.92727i 2.21195 − 0.327549i −1.86112 + 1.41526i 1.00000i 1.04816 + 2.62704i −0.266637 −2.20962 + 2.26221i
29.11 −0.752500 − 1.19739i −2.12140 −0.867487 + 1.80207i −2.02293 + 0.952769i 1.59636 + 2.54015i 1.00000i 2.81057 − 0.317339i 1.50036 2.66309 + 1.70527i
29.12 −0.752500 + 1.19739i −2.12140 −0.867487 − 1.80207i −2.02293 − 0.952769i 1.59636 − 2.54015i − 1.00000i 2.81057 + 0.317339i 1.50036 2.66309 − 1.70527i
29.13 −0.614166 − 1.27389i −2.96656 −1.24560 + 1.56476i 2.18846 + 0.458941i 1.82196 + 3.77907i − 1.00000i 2.75834 + 0.625738i 5.80046 −0.759437 − 3.06973i
29.14 −0.614166 + 1.27389i −2.96656 −1.24560 − 1.56476i 2.18846 − 0.458941i 1.82196 − 3.77907i 1.00000i 2.75834 − 0.625738i 5.80046 −0.759437 + 3.06973i
29.15 −0.269098 − 1.38838i 2.55593 −1.85517 + 0.747219i 0.790907 + 2.09152i −0.687797 − 3.54859i 1.00000i 1.53664 + 2.37460i 3.53279 2.69099 − 1.66090i
29.16 −0.269098 + 1.38838i 2.55593 −1.85517 − 0.747219i 0.790907 − 2.09152i −0.687797 + 3.54859i − 1.00000i 1.53664 − 2.37460i 3.53279 2.69099 + 1.66090i
29.17 −0.139020 − 1.40736i −0.319826 −1.96135 + 0.391304i −1.20181 + 1.88564i 0.0444622 + 0.450112i − 1.00000i 0.823373 + 2.70593i −2.89771 2.82086 + 1.42925i
29.18 −0.139020 + 1.40736i −0.319826 −1.96135 − 0.391304i −1.20181 − 1.88564i 0.0444622 − 0.450112i 1.00000i 0.823373 − 2.70593i −2.89771 2.82086 − 1.42925i
29.19 0.139020 − 1.40736i 0.319826 −1.96135 − 0.391304i 1.20181 − 1.88564i 0.0444622 − 0.450112i − 1.00000i −0.823373 + 2.70593i −2.89771 −2.48671 − 1.95353i
29.20 0.139020 + 1.40736i 0.319826 −1.96135 + 0.391304i 1.20181 + 1.88564i 0.0444622 + 0.450112i 1.00000i −0.823373 − 2.70593i −2.89771 −2.48671 + 1.95353i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.l.a ✓ 36
4.b odd 2 1 1120.2.l.a 36
5.b even 2 1 inner 280.2.l.a ✓ 36
8.b even 2 1 inner 280.2.l.a ✓ 36
8.d odd 2 1 1120.2.l.a 36
20.d odd 2 1 1120.2.l.a 36
40.e odd 2 1 1120.2.l.a 36
40.f even 2 1 inner 280.2.l.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.l.a ✓ 36 1.a even 1 1 trivial
280.2.l.a ✓ 36 5.b even 2 1 inner
280.2.l.a ✓ 36 8.b even 2 1 inner
280.2.l.a ✓ 36 40.f even 2 1 inner
1120.2.l.a 36 4.b odd 2 1
1120.2.l.a 36 8.d odd 2 1
1120.2.l.a 36 20.d odd 2 1
1120.2.l.a 36 40.e odd 2 1

Hecke kernels

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