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Properties

Label 140.3.f.a
Level $140$
Weight $3$
Character orbit 140.f
Analytic conductor $3.815$
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] = [140,3,Mod(99,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 140.f (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: \(3.81472370104\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{4} + 4 q^{5} + 12 q^{6} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{4} + 4 q^{5} + 12 q^{6} + 108 q^{9} - 8 q^{10} - 108 q^{16} + 52 q^{20} + 88 q^{24} - 60 q^{25} + 148 q^{26} - 40 q^{29} - 180 q^{30} - 28 q^{34} - 260 q^{36} + 44 q^{40} - 200 q^{41} - 16 q^{44} + 108 q^{45} + 96 q^{46} + 252 q^{49} - 156 q^{50} - 292 q^{54} - 84 q^{56} - 44 q^{60} + 136 q^{61} + 188 q^{64} - 240 q^{65} - 148 q^{66} - 672 q^{69} - 84 q^{70} + 304 q^{74} + 528 q^{76} + 108 q^{80} + 164 q^{81} + 192 q^{85} - 144 q^{86} + 520 q^{89} + 772 q^{90} + 212 q^{94} - 240 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −1.95169 − 0.436922i −3.00385 3.61820 + 1.70547i −2.29110 − 4.44420i 5.86259 + 1.31245i 2.64575 −6.31645 − 4.90943i 0.0231246 2.52975 + 9.67473i
99.2 −1.95169 + 0.436922i −3.00385 3.61820 − 1.70547i −2.29110 + 4.44420i 5.86259 − 1.31245i 2.64575 −6.31645 + 4.90943i 0.0231246 2.52975 − 9.67473i
99.3 −1.90238 − 0.617206i −0.587922 3.23811 + 2.34832i 3.06497 + 3.95044i 1.11845 + 0.362869i −2.64575 −4.71073 − 6.46599i −8.65435 −3.39250 − 9.40696i
99.4 −1.90238 + 0.617206i −0.587922 3.23811 − 2.34832i 3.06497 − 3.95044i 1.11845 − 0.362869i −2.64575 −4.71073 + 6.46599i −8.65435 −3.39250 + 9.40696i
99.5 −1.78382 − 0.904428i 3.77757 2.36402 + 3.22667i 4.48346 − 2.21328i −6.73851 − 3.41654i 2.64575 −1.29869 − 7.89388i 5.27006 −9.99943 − 0.106865i
99.6 −1.78382 + 0.904428i 3.77757 2.36402 − 3.22667i 4.48346 + 2.21328i −6.73851 + 3.41654i 2.64575 −1.29869 + 7.89388i 5.27006 −9.99943 + 0.106865i
99.7 −1.41922 − 1.40919i −4.84964 0.0283490 + 3.99990i −4.06042 + 2.91770i 6.88269 + 6.83408i −2.64575 5.59640 − 5.71667i 14.5190 9.87422 + 1.58106i
99.8 −1.41922 + 1.40919i −4.84964 0.0283490 − 3.99990i −4.06042 − 2.91770i 6.88269 − 6.83408i −2.64575 5.59640 + 5.71667i 14.5190 9.87422 − 1.58106i
99.9 −1.38254 − 1.44519i 1.60492 −0.177168 + 3.99607i −2.32719 − 4.42540i −2.21887 − 2.31943i −2.64575 6.02004 − 5.26869i −6.42422 −3.17813 + 9.48153i
99.10 −1.38254 + 1.44519i 1.60492 −0.177168 − 3.99607i −2.32719 + 4.42540i −2.21887 + 2.31943i −2.64575 6.02004 + 5.26869i −6.42422 −3.17813 − 9.48153i
99.11 −1.17237 − 1.62035i −3.33214 −1.25109 + 3.79931i 4.99443 − 0.235983i 3.90651 + 5.39924i 2.64575 7.62297 − 2.42701i 2.10314 −6.23771 − 7.81607i
99.12 −1.17237 + 1.62035i −3.33214 −1.25109 − 3.79931i 4.99443 + 0.235983i 3.90651 − 5.39924i 2.64575 7.62297 + 2.42701i 2.10314 −6.23771 + 7.81607i
99.13 −0.961741 − 1.75358i 5.70307 −2.15011 + 3.37299i 0.771691 + 4.94009i −5.48487 − 10.0008i −2.64575 7.98266 + 0.526460i 23.5250 7.92069 − 6.10431i
99.14 −0.961741 + 1.75358i 5.70307 −2.15011 − 3.37299i 0.771691 − 4.94009i −5.48487 + 10.0008i −2.64575 7.98266 − 0.526460i 23.5250 7.92069 + 6.10431i
99.15 −0.813487 − 1.82708i 0.660255 −2.67648 + 2.97262i −4.66425 + 1.80133i −0.537109 − 1.20634i 2.64575 7.60851 + 2.47196i −8.56406 7.08549 + 7.05662i
99.16 −0.813487 + 1.82708i 0.660255 −2.67648 − 2.97262i −4.66425 − 1.80133i −0.537109 + 1.20634i 2.64575 7.60851 − 2.47196i −8.56406 7.08549 − 7.05662i
99.17 −0.0554918 − 1.99923i −3.76859 −3.99384 + 0.221882i 1.02841 − 4.89309i 0.209126 + 7.53429i −2.64575 0.665218 + 7.97229i 5.20230 −9.83949 − 1.78451i
99.18 −0.0554918 + 1.99923i −3.76859 −3.99384 − 0.221882i 1.02841 + 4.89309i 0.209126 − 7.53429i −2.64575 0.665218 − 7.97229i 5.20230 −9.83949 + 1.78451i
99.19 0.0554918 − 1.99923i 3.76859 −3.99384 − 0.221882i 1.02841 − 4.89309i 0.209126 − 7.53429i 2.64575 −0.665218 + 7.97229i 5.20230 −9.72535 − 2.32756i
99.20 0.0554918 + 1.99923i 3.76859 −3.99384 + 0.221882i 1.02841 + 4.89309i 0.209126 + 7.53429i 2.64575 −0.665218 − 7.97229i 5.20230 −9.72535 + 2.32756i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.3.f.a ✓ 36
4.b odd 2 1 inner 140.3.f.a ✓ 36
5.b even 2 1 inner 140.3.f.a ✓ 36
20.d odd 2 1 inner 140.3.f.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.3.f.a ✓ 36 1.a even 1 1 trivial
140.3.f.a ✓ 36 4.b odd 2 1 inner
140.3.f.a ✓ 36 5.b even 2 1 inner
140.3.f.a ✓ 36 20.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(140, [\chi])\).