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Properties

Label 112.2.v.a
Level $112$
Weight $2$
Character orbit 112.v
Analytic conductor $0.894$
Analytic rank $0$
Dimension $56$
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] = [112,2,Mod(3,112)] 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("112.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.v (of order \(12\), degree \(4\), 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: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{2} - 6 q^{3} - 4 q^{4} - 6 q^{5} - 8 q^{7} + 4 q^{8} - 24 q^{10} + 2 q^{11} - 6 q^{12} + 16 q^{14} + 8 q^{16} - 12 q^{17} - 30 q^{18} - 6 q^{19} - 10 q^{21} - 28 q^{22} - 12 q^{23} - 6 q^{24}+ \cdots - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41419 − 0.00846089i 1.95777 − 0.524583i 1.99986 + 0.0239306i 0.666898 − 2.48890i −2.77309 + 0.725295i −2.38027 + 1.15512i −2.82797 − 0.0507629i 0.959602 − 0.554026i −0.964178 + 3.51413i
3.2 −1.25731 − 0.647437i −3.08091 + 0.825526i 1.16165 + 1.62806i 0.501060 − 1.86998i 4.40813 + 0.956753i 2.17953 + 1.49988i −0.406487 − 2.79907i 6.21241 − 3.58674i −1.84068 + 2.02674i
3.3 −1.25445 + 0.652965i −1.05433 + 0.282507i 1.14727 − 1.63822i −0.374919 + 1.39922i 1.13814 − 1.04283i −0.298614 + 2.62885i −0.369496 + 2.80419i −1.56627 + 0.904287i −0.443324 − 2.00005i
3.4 −0.946521 + 1.05076i 2.77508 − 0.743580i −0.208197 − 1.98913i −0.907491 + 3.38680i −1.84535 + 3.61976i 0.0791552 − 2.64457i 2.28717 + 1.66399i 4.55008 − 2.62699i −2.69976 − 4.15923i
3.5 −0.910025 − 1.08252i 1.67893 − 0.449868i −0.343708 + 1.97024i 0.195879 − 0.731029i −2.01486 − 1.40809i 2.52163 − 0.800849i 2.44562 − 1.42090i 0.0183525 − 0.0105958i −0.969610 + 0.453212i
3.6 −0.799555 − 1.16650i −0.763582 + 0.204601i −0.721425 + 1.86535i −1.02188 + 3.81370i 0.849192 + 0.727125i −2.64575 + 0.00379639i 2.75275 − 0.649913i −2.05688 + 1.18754i 5.26571 − 1.85724i
3.7 −0.252874 + 1.39142i −2.61463 + 0.700587i −1.87211 − 0.703710i 0.120751 − 0.450647i −0.313640 − 3.81521i −2.37326 − 1.16946i 1.45257 − 2.42694i 3.74737 − 2.16355i 0.596506 + 0.281972i
3.8 0.149759 − 1.40626i −1.44888 + 0.388227i −1.95514 − 0.421199i 0.755106 − 2.81809i 0.328966 + 2.09565i −1.47726 − 2.19493i −0.885116 + 2.68637i −0.649537 + 0.375010i −3.84990 − 1.48391i
3.9 0.187641 + 1.40171i 0.198087 − 0.0530773i −1.92958 + 0.526037i −0.487744 + 1.82029i 0.111568 + 0.267701i 1.84933 + 1.89208i −1.09942 − 2.60601i −2.56165 + 1.47897i −2.64303 − 0.342115i
3.10 0.438321 − 1.34457i 2.29647 − 0.615336i −1.61575 − 1.17871i −0.223959 + 0.835825i 0.179226 − 3.35748i −1.25761 + 2.32775i −2.29308 + 1.65584i 2.29704 − 1.32620i 1.02566 + 0.667489i
3.11 0.898888 + 1.09179i 1.30734 − 0.350301i −0.384000 + 1.96279i 0.294400 − 1.09872i 1.55761 + 1.11246i −1.56831 − 2.13082i −2.48812 + 1.34508i −1.01165 + 0.584076i 1.46420 − 0.666200i
3.12 1.06970 − 0.925063i −0.246909 + 0.0661591i 0.288515 − 1.97908i −0.133797 + 0.499339i −0.202917 + 0.299177i 2.45011 − 0.998472i −1.52215 − 2.38392i −2.54149 + 1.46733i 0.318797 + 0.657913i
3.13 1.31418 + 0.522432i −2.48480 + 0.665801i 1.45413 + 1.37314i −0.837076 + 3.12401i −3.61331 − 0.423158i 1.56215 − 2.13534i 1.19362 + 2.56423i 3.13287 − 1.80877i −2.73215 + 3.66819i
3.14 1.41041 + 0.103712i −0.885661 + 0.237312i 1.97849 + 0.292553i 0.818796 − 3.05579i −1.27375 + 0.242853i −0.640837 + 2.56697i 2.76013 + 0.617811i −1.87000 + 1.07964i 1.47176 − 4.22498i
19.1 −1.38359 + 0.292689i 0.615336 − 2.29647i 1.82867 − 0.809925i −0.835825 + 0.223959i −0.179226 + 3.35748i −1.25761 − 2.32775i −2.29308 + 1.65584i −2.29704 − 1.32620i 1.09089 − 0.554505i
19.2 −1.33598 − 0.463855i −0.0661591 + 0.246909i 1.56968 + 1.23940i −0.499339 + 0.133797i 0.202917 − 0.299177i 2.45011 + 0.998472i −1.52215 − 2.38392i 2.54149 + 1.46733i 0.729168 + 0.0528705i
19.3 −1.29274 + 0.573436i −0.388227 + 1.44888i 1.34234 − 1.48261i 2.81809 − 0.755106i −0.328966 − 2.09565i −1.47726 + 2.19493i −0.885116 + 2.68637i 0.649537 + 0.375010i −3.21005 + 2.59215i
19.4 −0.615385 − 1.27330i −0.237312 + 0.885661i −1.24260 + 1.56714i 3.05579 − 0.818796i 1.27375 − 0.242853i −0.640837 − 2.56697i 2.76013 + 0.617811i 1.87000 + 1.07964i −2.92306 − 3.38707i
19.5 −0.610438 + 1.27568i −0.204601 + 0.763582i −1.25473 − 1.55745i −3.81370 + 1.02188i −0.849192 − 0.727125i −2.64575 − 0.00379639i 2.75275 − 0.649913i 2.05688 + 1.18754i 1.02443 − 5.48886i
19.6 −0.482479 + 1.32937i 0.449868 − 1.67893i −1.53443 − 1.28278i 0.731029 − 0.195879i 2.01486 + 1.40809i 2.52163 + 0.800849i 2.44562 − 1.42090i −0.0183525 − 0.0105958i −0.0923119 + 1.06631i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
16.f odd 4 1 inner
112.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.v.a ✓ 56
4.b odd 2 1 448.2.z.a 56
7.b odd 2 1 784.2.w.f 56
7.c even 3 1 784.2.j.a 56
7.c even 3 1 784.2.w.f 56
7.d odd 6 1 inner 112.2.v.a ✓ 56
7.d odd 6 1 784.2.j.a 56
8.b even 2 1 896.2.z.b 56
8.d odd 2 1 896.2.z.a 56
16.e even 4 1 448.2.z.a 56
16.e even 4 1 896.2.z.a 56
16.f odd 4 1 inner 112.2.v.a ✓ 56
16.f odd 4 1 896.2.z.b 56
28.f even 6 1 448.2.z.a 56
56.j odd 6 1 896.2.z.b 56
56.m even 6 1 896.2.z.a 56
112.j even 4 1 784.2.w.f 56
112.u odd 12 1 784.2.j.a 56
112.u odd 12 1 784.2.w.f 56
112.v even 12 1 inner 112.2.v.a ✓ 56
112.v even 12 1 784.2.j.a 56
112.v even 12 1 896.2.z.b 56
112.x odd 12 1 448.2.z.a 56
112.x odd 12 1 896.2.z.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.v.a ✓ 56 1.a even 1 1 trivial
112.2.v.a ✓ 56 7.d odd 6 1 inner
112.2.v.a ✓ 56 16.f odd 4 1 inner
112.2.v.a ✓ 56 112.v even 12 1 inner
448.2.z.a 56 4.b odd 2 1
448.2.z.a 56 16.e even 4 1
448.2.z.a 56 28.f even 6 1
448.2.z.a 56 112.x odd 12 1
784.2.j.a 56 7.c even 3 1
784.2.j.a 56 7.d odd 6 1
784.2.j.a 56 112.u odd 12 1
784.2.j.a 56 112.v even 12 1
784.2.w.f 56 7.b odd 2 1
784.2.w.f 56 7.c even 3 1
784.2.w.f 56 112.j even 4 1
784.2.w.f 56 112.u odd 12 1
896.2.z.a 56 8.d odd 2 1
896.2.z.a 56 16.e even 4 1
896.2.z.a 56 56.m even 6 1
896.2.z.a 56 112.x odd 12 1
896.2.z.b 56 8.b even 2 1
896.2.z.b 56 16.f odd 4 1
896.2.z.b 56 56.j odd 6 1
896.2.z.b 56 112.v even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(112, [\chi])\).