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Properties

Label 322.2.k.a
Level $322$
Weight $2$
Character orbit 322.k
Analytic conductor $2.571$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,2,Mod(83,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.83");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.k (of order \(22\), degree \(10\), 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: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 8 q^{2} - 8 q^{4} - 11 q^{7} - 8 q^{8} - 6 q^{9} + 11 q^{14} - 8 q^{16} - 6 q^{18} - 33 q^{21} + 18 q^{23} - 58 q^{25} - 11 q^{28} + 16 q^{29} + 22 q^{30} - 8 q^{32} + 47 q^{35} + 16 q^{36} - 22 q^{37}+ \cdots - 44 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −0.959493 + 0.281733i −1.73925 + 1.50707i 0.841254 − 0.540641i 0.406529 + 2.82747i 1.24421 − 1.93603i 0.697541 + 2.55214i −0.654861 + 0.755750i 0.326790 − 2.27288i −1.18665 − 2.59840i
83.2 −0.959493 + 0.281733i −1.70070 + 1.47367i 0.841254 − 0.540641i 0.0506866 + 0.352533i 1.21663 − 1.89312i −2.46109 − 0.971103i −0.654861 + 0.755750i 0.293752 − 2.04309i −0.147953 − 0.323973i
83.3 −0.959493 + 0.281733i −0.749737 + 0.649651i 0.841254 − 0.540641i −0.540672 − 3.76045i 0.536339 − 0.834560i −2.61856 − 0.378356i −0.654861 + 0.755750i −0.286885 + 1.99533i 1.57821 + 3.45580i
83.4 −0.959493 + 0.281733i −0.198408 + 0.171921i 0.841254 − 0.540641i −0.382697 − 2.66172i 0.141935 − 0.220855i 0.921868 + 2.47995i −0.654861 + 0.755750i −0.417136 + 2.90124i 1.11709 + 2.44608i
83.5 −0.959493 + 0.281733i 0.198408 − 0.171921i 0.841254 − 0.540641i 0.382697 + 2.66172i −0.141935 + 0.220855i 2.47792 − 0.927322i −0.654861 + 0.755750i −0.417136 + 2.90124i −1.11709 − 2.44608i
83.6 −0.959493 + 0.281733i 0.749737 − 0.649651i 0.841254 − 0.540641i 0.540672 + 3.76045i −0.536339 + 0.834560i −2.00073 − 1.73120i −0.654861 + 0.755750i −0.286885 + 1.99533i −1.57821 − 3.45580i
83.7 −0.959493 + 0.281733i 1.70070 − 1.47367i 0.841254 − 0.540641i −0.0506866 − 0.352533i −1.21663 + 1.89312i −2.34558 − 1.22403i −0.654861 + 0.755750i 0.293752 − 2.04309i 0.147953 + 0.323973i
83.8 −0.959493 + 0.281733i 1.73925 − 1.50707i 0.841254 − 0.540641i −0.406529 − 2.82747i −1.24421 + 1.93603i 2.38557 − 1.14413i −0.654861 + 0.755750i 0.326790 − 2.27288i 1.18665 + 2.59840i
97.1 −0.959493 − 0.281733i −1.73925 − 1.50707i 0.841254 + 0.540641i 0.406529 − 2.82747i 1.24421 + 1.93603i 0.697541 − 2.55214i −0.654861 − 0.755750i 0.326790 + 2.27288i −1.18665 + 2.59840i
97.2 −0.959493 − 0.281733i −1.70070 − 1.47367i 0.841254 + 0.540641i 0.0506866 − 0.352533i 1.21663 + 1.89312i −2.46109 + 0.971103i −0.654861 − 0.755750i 0.293752 + 2.04309i −0.147953 + 0.323973i
97.3 −0.959493 − 0.281733i −0.749737 − 0.649651i 0.841254 + 0.540641i −0.540672 + 3.76045i 0.536339 + 0.834560i −2.61856 + 0.378356i −0.654861 − 0.755750i −0.286885 − 1.99533i 1.57821 − 3.45580i
97.4 −0.959493 − 0.281733i −0.198408 − 0.171921i 0.841254 + 0.540641i −0.382697 + 2.66172i 0.141935 + 0.220855i 0.921868 − 2.47995i −0.654861 − 0.755750i −0.417136 − 2.90124i 1.11709 − 2.44608i
97.5 −0.959493 − 0.281733i 0.198408 + 0.171921i 0.841254 + 0.540641i 0.382697 − 2.66172i −0.141935 − 0.220855i 2.47792 + 0.927322i −0.654861 − 0.755750i −0.417136 − 2.90124i −1.11709 + 2.44608i
97.6 −0.959493 − 0.281733i 0.749737 + 0.649651i 0.841254 + 0.540641i 0.540672 − 3.76045i −0.536339 − 0.834560i −2.00073 + 1.73120i −0.654861 − 0.755750i −0.286885 − 1.99533i −1.57821 + 3.45580i
97.7 −0.959493 − 0.281733i 1.70070 + 1.47367i 0.841254 + 0.540641i −0.0506866 + 0.352533i −1.21663 − 1.89312i −2.34558 + 1.22403i −0.654861 − 0.755750i 0.293752 + 2.04309i 0.147953 − 0.323973i
97.8 −0.959493 − 0.281733i 1.73925 + 1.50707i 0.841254 + 0.540641i −0.406529 + 2.82747i −1.24421 − 1.93603i 2.38557 + 1.14413i −0.654861 − 0.755750i 0.326790 + 2.27288i 1.18665 − 2.59840i
111.1 0.415415 + 0.909632i −0.760851 − 2.59122i −0.654861 + 0.755750i 2.71881 + 1.74727i 2.04099 − 1.76853i 0.0953475 − 2.64403i −0.959493 − 0.281733i −3.61176 + 2.32114i −0.459941 + 3.19896i
111.2 0.415415 + 0.909632i −0.730873 − 2.48912i −0.654861 + 0.755750i −1.52995 − 0.983241i 1.96057 − 1.69884i −2.20702 + 1.45913i −0.959493 − 0.281733i −3.13780 + 2.01654i 0.258822 − 1.80015i
111.3 0.415415 + 0.909632i −0.229469 − 0.781499i −0.654861 + 0.755750i −1.74460 − 1.12119i 0.615551 − 0.533378i −0.251104 − 2.63381i −0.959493 − 0.281733i 1.96568 − 1.26326i 0.295134 − 2.05270i
111.4 0.415415 + 0.909632i −0.192661 − 0.656142i −0.654861 + 0.755750i −2.27924 − 1.46478i 0.516813 − 0.447821i 2.42953 + 1.04756i −0.959493 − 0.281733i 2.13036 − 1.36910i 0.385580 − 2.68177i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.d odd 22 1 inner
161.k even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.k.a ✓ 80
7.b odd 2 1 inner 322.2.k.a ✓ 80
23.d odd 22 1 inner 322.2.k.a ✓ 80
161.k even 22 1 inner 322.2.k.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.k.a ✓ 80 1.a even 1 1 trivial
322.2.k.a ✓ 80 7.b odd 2 1 inner
322.2.k.a ✓ 80 23.d odd 22 1 inner
322.2.k.a ✓ 80 161.k even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 9 T_{3}^{78} + 119 T_{3}^{76} - 1342 T_{3}^{74} + 17316 T_{3}^{72} - 101532 T_{3}^{70} + \cdots + 956720690641 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display