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Properties

Label 140.2.k.a
Level $140$
Weight $2$
Character orbit 140.k
Analytic conductor $1.118$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [140,2,Mod(43,140)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("140.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(140, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{6} - 16 q^{10} - 16 q^{12} - 4 q^{13} - 8 q^{16} - 20 q^{17} + 28 q^{18} + 20 q^{20} + 4 q^{22} - 20 q^{25} - 32 q^{26} - 4 q^{30} + 20 q^{37} - 36 q^{40} - 20 q^{42} + 20 q^{45} + 16 q^{46}+ \cdots + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.38250 − 0.297828i −0.137886 + 0.137886i 1.82260 + 0.823494i −0.499053 + 2.17967i 0.231693 − 0.149560i −0.707107 − 0.707107i −2.27447 − 1.68130i 2.96198i 1.33911 − 2.86475i
43.2 −1.36728 + 0.361308i −1.26588 + 1.26588i 1.73891 − 0.988019i 1.96164 − 1.07330i 1.27344 − 2.18818i 0.707107 + 0.707107i −2.02060 + 1.97918i − 0.204893i −2.29431 + 2.17626i
43.3 −1.25629 + 0.649412i 2.28163 − 2.28163i 1.15653 − 1.63170i 0.0430826 + 2.23565i −1.38467 + 4.34811i 0.707107 + 0.707107i −0.393286 + 2.80095i − 7.41170i −1.50599 − 2.78065i
43.4 −1.16481 − 0.802007i −1.75731 + 1.75731i 0.713568 + 1.86837i −0.854664 − 2.06629i 3.45630 − 0.637557i −0.707107 − 0.707107i 0.667278 − 2.74859i − 3.17626i −0.661657 + 3.09228i
43.5 −1.08834 − 0.903055i 1.00798 − 1.00798i 0.368985 + 1.96567i 2.21855 − 0.279354i −2.00729 + 0.186768i 0.707107 + 0.707107i 1.37352 − 2.47254i 0.967954i −2.66682 − 1.69944i
43.6 −0.649412 + 1.25629i −2.28163 + 2.28163i −1.15653 − 1.63170i 0.0430826 + 2.23565i −1.38467 − 4.34811i −0.707107 − 0.707107i 2.80095 − 0.393286i − 7.41170i −2.83661 − 1.39774i
43.7 −0.622342 − 1.26992i 2.09607 − 2.09607i −1.22538 + 1.58065i −2.14272 − 0.639344i −3.96631 − 1.35737i −0.707107 − 0.707107i 2.76990 + 0.572433i − 5.78704i 0.521588 + 3.11897i
43.8 −0.396294 − 1.35755i −0.945787 + 0.945787i −1.68590 + 1.07598i −1.94751 + 1.09873i 1.65877 + 0.909146i 0.707107 + 0.707107i 2.12881 + 1.86230i 1.21097i 2.26337 + 2.20843i
43.9 −0.361308 + 1.36728i 1.26588 − 1.26588i −1.73891 − 0.988019i 1.96164 − 1.07330i 1.27344 + 2.18818i −0.707107 − 0.707107i 1.97918 − 2.02060i − 0.204893i 0.758752 + 3.06990i
43.10 0.297828 + 1.38250i 0.137886 − 0.137886i −1.82260 + 0.823494i −0.499053 + 2.17967i 0.231693 + 0.149560i 0.707107 + 0.707107i −1.68130 − 2.27447i 2.96198i −3.16201 − 0.0407730i
43.11 0.447909 − 1.34141i 0.396892 − 0.396892i −1.59875 − 1.20166i −0.137858 − 2.23181i −0.354623 − 0.710167i 0.707107 + 0.707107i −2.32801 + 1.60635i 2.68495i −3.05552 − 0.814726i
43.12 0.578354 − 1.29055i 1.27396 − 1.27396i −1.33101 − 1.49278i 1.35854 + 1.77606i −0.907305 − 2.38090i −0.707107 − 0.707107i −2.69630 + 0.854378i − 0.245954i 3.07780 − 0.726061i
43.13 0.802007 + 1.16481i 1.75731 − 1.75731i −0.713568 + 1.86837i −0.854664 − 2.06629i 3.45630 + 0.637557i 0.707107 + 0.707107i −2.74859 + 0.667278i − 3.17626i 1.72139 − 2.65270i
43.14 0.903055 + 1.08834i −1.00798 + 1.00798i −0.368985 + 1.96567i 2.21855 − 0.279354i −2.00729 − 0.186768i −0.707107 − 0.707107i −2.47254 + 1.37352i 0.967954i 2.30750 + 2.16227i
43.15 1.26992 + 0.622342i −2.09607 + 2.09607i 1.22538 + 1.58065i −2.14272 − 0.639344i −3.96631 + 1.35737i 0.707107 + 0.707107i 0.572433 + 2.76990i − 5.78704i −2.32318 − 2.14542i
43.16 1.29055 − 0.578354i −1.27396 + 1.27396i 1.33101 − 1.49278i 1.35854 + 1.77606i −0.907305 + 2.38090i 0.707107 + 0.707107i 0.854378 − 2.69630i − 0.245954i 2.78044 + 1.50637i
43.17 1.34141 − 0.447909i −0.396892 + 0.396892i 1.59875 − 1.20166i −0.137858 − 2.23181i −0.354623 + 0.710167i −0.707107 − 0.707107i 1.60635 − 2.32801i 2.68495i −1.18457 − 2.93203i
43.18 1.35755 + 0.396294i 0.945787 − 0.945787i 1.68590 + 1.07598i −1.94751 + 1.09873i 1.65877 − 0.909146i −0.707107 − 0.707107i 1.86230 + 2.12881i 1.21097i −3.07927 + 0.719794i
127.1 −1.38250 + 0.297828i −0.137886 − 0.137886i 1.82260 − 0.823494i −0.499053 − 2.17967i 0.231693 + 0.149560i −0.707107 + 0.707107i −2.27447 + 1.68130i − 2.96198i 1.33911 + 2.86475i
127.2 −1.36728 − 0.361308i −1.26588 − 1.26588i 1.73891 + 0.988019i 1.96164 + 1.07330i 1.27344 + 2.18818i 0.707107 − 0.707107i −2.02060 − 1.97918i 0.204893i −2.29431 − 2.17626i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.18
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.k.a ✓ 36
4.b odd 2 1 inner 140.2.k.a ✓ 36
5.b even 2 1 700.2.k.b 36
5.c odd 4 1 inner 140.2.k.a ✓ 36
5.c odd 4 1 700.2.k.b 36
7.b odd 2 1 980.2.k.l 36
7.c even 3 2 980.2.x.k 72
7.d odd 6 2 980.2.x.l 72
20.d odd 2 1 700.2.k.b 36
20.e even 4 1 inner 140.2.k.a ✓ 36
20.e even 4 1 700.2.k.b 36
28.d even 2 1 980.2.k.l 36
28.f even 6 2 980.2.x.l 72
28.g odd 6 2 980.2.x.k 72
35.f even 4 1 980.2.k.l 36
35.k even 12 2 980.2.x.l 72
35.l odd 12 2 980.2.x.k 72
140.j odd 4 1 980.2.k.l 36
140.w even 12 2 980.2.x.k 72
140.x odd 12 2 980.2.x.l 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.k.a ✓ 36 1.a even 1 1 trivial
140.2.k.a ✓ 36 4.b odd 2 1 inner
140.2.k.a ✓ 36 5.c odd 4 1 inner
140.2.k.a ✓ 36 20.e even 4 1 inner
700.2.k.b 36 5.b even 2 1
700.2.k.b 36 5.c odd 4 1
700.2.k.b 36 20.d odd 2 1
700.2.k.b 36 20.e even 4 1
980.2.k.l 36 7.b odd 2 1
980.2.k.l 36 28.d even 2 1
980.2.k.l 36 35.f even 4 1
980.2.k.l 36 140.j odd 4 1
980.2.x.k 72 7.c even 3 2
980.2.x.k 72 28.g odd 6 2
980.2.x.k 72 35.l odd 12 2
980.2.x.k 72 140.w even 12 2
980.2.x.l 72 7.d odd 6 2
980.2.x.l 72 28.f even 6 2
980.2.x.l 72 35.k even 12 2
980.2.x.l 72 140.x odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(140, [\chi])\).