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Properties

Label 322.2.k.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{2} - 8 q^{4} + 11 q^{7} + 8 q^{8} + 14 q^{9} - 11 q^{14} - 8 q^{16} - 14 q^{18} - 33 q^{21} + 18 q^{23} + 6 q^{25} - 11 q^{28} + 8 q^{29} + 22 q^{30} + 8 q^{32} - 33 q^{35} - 8 q^{36} - 22 q^{37}+ \cdots + 132 q^{99}+O(q^{100}) \) 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 −2.27797 + 1.97387i 0.841254 − 0.540641i −0.347463 − 2.41666i −1.62959 + 2.53570i −2.04477 − 1.67896i 0.654861 − 0.755750i 0.866033 − 6.02339i −1.01424 − 2.22088i
83.2 0.959493 − 0.281733i −1.76160 + 1.52644i 0.841254 − 0.540641i 0.360367 + 2.50641i −1.26020 + 1.96091i 1.49769 − 2.18104i 0.654861 − 0.755750i 0.346290 − 2.40850i 1.05191 + 2.30335i
83.3 0.959493 − 0.281733i −1.04759 + 0.907744i 0.841254 − 0.540641i −0.229109 − 1.59349i −0.749417 + 1.16612i 2.48964 + 0.895380i 0.654861 − 0.755750i −0.153494 + 1.06757i −0.668765 − 1.46439i
83.4 0.959493 − 0.281733i −0.668109 + 0.578920i 0.841254 − 0.540641i 0.308243 + 2.14388i −0.477945 + 0.743697i −1.85023 + 1.89121i 0.654861 − 0.755750i −0.315723 + 2.19590i 0.899756 + 1.97019i
83.5 0.959493 − 0.281733i 0.668109 − 0.578920i 0.841254 − 0.540641i −0.308243 − 2.14388i 0.477945 − 0.743697i 0.217640 − 2.63678i 0.654861 − 0.755750i −0.315723 + 2.19590i −0.899756 − 1.97019i
83.6 0.959493 − 0.281733i 1.04759 − 0.907744i 0.841254 − 0.540641i 0.229109 + 1.59349i 0.749417 − 1.16612i 2.30705 + 1.29519i 0.654861 − 0.755750i −0.153494 + 1.06757i 0.668765 + 1.46439i
83.7 0.959493 − 0.281733i 1.76160 − 1.52644i 0.841254 − 0.540641i −0.360367 − 2.50641i 1.26020 − 1.96091i −0.667538 + 2.56015i 0.654861 − 0.755750i 0.346290 − 2.40850i −1.05191 − 2.30335i
83.8 0.959493 − 0.281733i 2.27797 − 1.97387i 0.841254 − 0.540641i 0.347463 + 2.41666i 1.62959 − 2.53570i −2.60791 − 0.445849i 0.654861 − 0.755750i 0.866033 − 6.02339i 1.01424 + 2.22088i
97.1 0.959493 + 0.281733i −2.27797 − 1.97387i 0.841254 + 0.540641i −0.347463 + 2.41666i −1.62959 − 2.53570i −2.04477 + 1.67896i 0.654861 + 0.755750i 0.866033 + 6.02339i −1.01424 + 2.22088i
97.2 0.959493 + 0.281733i −1.76160 − 1.52644i 0.841254 + 0.540641i 0.360367 − 2.50641i −1.26020 − 1.96091i 1.49769 + 2.18104i 0.654861 + 0.755750i 0.346290 + 2.40850i 1.05191 − 2.30335i
97.3 0.959493 + 0.281733i −1.04759 − 0.907744i 0.841254 + 0.540641i −0.229109 + 1.59349i −0.749417 − 1.16612i 2.48964 − 0.895380i 0.654861 + 0.755750i −0.153494 − 1.06757i −0.668765 + 1.46439i
97.4 0.959493 + 0.281733i −0.668109 − 0.578920i 0.841254 + 0.540641i 0.308243 − 2.14388i −0.477945 − 0.743697i −1.85023 − 1.89121i 0.654861 + 0.755750i −0.315723 − 2.19590i 0.899756 − 1.97019i
97.5 0.959493 + 0.281733i 0.668109 + 0.578920i 0.841254 + 0.540641i −0.308243 + 2.14388i 0.477945 + 0.743697i 0.217640 + 2.63678i 0.654861 + 0.755750i −0.315723 − 2.19590i −0.899756 + 1.97019i
97.6 0.959493 + 0.281733i 1.04759 + 0.907744i 0.841254 + 0.540641i 0.229109 − 1.59349i 0.749417 + 1.16612i 2.30705 − 1.29519i 0.654861 + 0.755750i −0.153494 − 1.06757i 0.668765 − 1.46439i
97.7 0.959493 + 0.281733i 1.76160 + 1.52644i 0.841254 + 0.540641i −0.360367 + 2.50641i 1.26020 + 1.96091i −0.667538 − 2.56015i 0.654861 + 0.755750i 0.346290 + 2.40850i −1.05191 + 2.30335i
97.8 0.959493 + 0.281733i 2.27797 + 1.97387i 0.841254 + 0.540641i 0.347463 − 2.41666i 1.62959 + 2.53570i −2.60791 + 0.445849i 0.654861 + 0.755750i 0.866033 + 6.02339i 1.01424 − 2.22088i
111.1 −0.415415 − 0.909632i −0.860482 − 2.93053i −0.654861 + 0.755750i 2.12917 + 1.36833i −2.30825 + 2.00011i −2.62625 − 0.320606i 0.959493 + 0.281733i −5.32384 + 3.42142i 0.360192 − 2.50519i
111.2 −0.415415 − 0.909632i −0.725956 − 2.47238i −0.654861 + 0.755750i −2.94826 − 1.89473i −1.94738 + 1.68742i 2.01573 − 1.71372i 0.959493 + 0.281733i −3.06188 + 1.96775i −0.498756 + 3.46893i
111.3 −0.415415 − 0.909632i −0.521769 − 1.77698i −0.654861 + 0.755750i 1.91138 + 1.22837i −1.39965 + 1.21280i 2.44569 + 1.00927i 0.959493 + 0.281733i −0.361658 + 0.232423i 0.323348 − 2.24893i
111.4 −0.415415 − 0.909632i −0.0730120 − 0.248656i −0.654861 + 0.755750i 0.853287 + 0.548374i −0.195855 + 0.169710i −1.00334 + 2.44812i 0.959493 + 0.281733i 2.46726 − 1.58561i 0.144351 − 1.00398i
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.b ✓ 80
7.b odd 2 1 inner 322.2.k.b ✓ 80
23.d odd 22 1 inner 322.2.k.b ✓ 80
161.k even 22 1 inner 322.2.k.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.k.b ✓ 80 1.a even 1 1 trivial
322.2.k.b ✓ 80 7.b odd 2 1 inner
322.2.k.b ✓ 80 23.d odd 22 1 inner
322.2.k.b ✓ 80 161.k even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 19 T_{3}^{78} + 183 T_{3}^{76} - 1172 T_{3}^{74} + 6522 T_{3}^{72} - 84094 T_{3}^{70} + \cdots + 19790033062801 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display