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Properties

Label 222.2.q.a
Level $222$
Weight $2$
Character orbit 222.q
Analytic conductor $1.773$
Analytic rank $0$
Dimension $144$
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] = [222,2,Mod(5,222)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(222, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 23]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("222.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 222.q (of order \(36\), degree \(12\), 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: \(1.77267892487\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(12\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 12 q^{9} - 12 q^{12} + 12 q^{15} - 36 q^{21} + 24 q^{28} - 168 q^{31} - 72 q^{34} - 24 q^{37} - 48 q^{40} + 12 q^{42} - 48 q^{43} - 48 q^{46} - 24 q^{49} + 36 q^{54} - 36 q^{57} - 24 q^{58} - 36 q^{63} + 24 q^{67} - 132 q^{69} + 24 q^{70} + 72 q^{75} - 48 q^{78} - 24 q^{79} - 156 q^{81} - 24 q^{82} - 72 q^{87} + 24 q^{88} - 120 q^{91} + 60 q^{93} + 144 q^{94} + 120 q^{97} + 156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.906308 − 0.422618i −1.60674 + 0.646820i 0.642788 + 0.766044i −0.198792 − 0.139196i 1.72956 + 0.0928210i −0.0256510 − 0.145474i −0.258819 − 0.965926i 2.16325 − 2.07855i 0.121340 + 0.210167i
5.2 −0.906308 − 0.422618i −1.35066 − 1.08431i 0.642788 + 0.766044i 1.61501 + 1.13085i 0.765862 + 1.55353i 0.209991 + 1.19092i −0.258819 − 0.965926i 0.648549 + 2.92906i −0.985784 − 1.70743i
5.3 −0.906308 − 0.422618i −0.460389 + 1.66974i 0.642788 + 0.766044i −2.23745 − 1.56668i 1.12292 − 1.31873i −0.378759 − 2.14805i −0.258819 − 0.965926i −2.57608 − 1.53746i 1.36571 + 2.36548i
5.4 −0.906308 − 0.422618i 0.121334 − 1.72780i 0.642788 + 0.766044i −1.16520 − 0.815881i −0.840164 + 1.51464i −0.192798 − 1.09341i −0.258819 − 0.965926i −2.97056 − 0.419280i 0.711222 + 1.23187i
5.5 −0.906308 − 0.422618i 1.45016 + 0.947126i 0.642788 + 0.766044i −0.535709 − 0.375107i −0.914015 − 1.47125i 0.567400 + 3.21788i −0.258819 − 0.965926i 1.20590 + 2.74696i 0.326990 + 0.566363i
5.6 −0.906308 − 0.422618i 1.72752 − 0.125233i 0.642788 + 0.766044i 1.40147 + 0.981317i −1.61859 − 0.616581i −0.864223 − 4.90125i −0.258819 − 0.965926i 2.96863 − 0.432685i −0.855437 − 1.48166i
5.7 0.906308 + 0.422618i −1.71968 − 0.206601i 0.642788 + 0.766044i 0.535709 + 0.375107i −1.47125 − 0.914015i 0.567400 + 3.21788i 0.258819 + 0.965926i 2.91463 + 0.710578i 0.326990 + 0.566363i
5.8 0.906308 + 0.422618i −1.24286 − 1.20636i 0.642788 + 0.766044i −1.40147 − 0.981317i −0.616581 − 1.61859i −0.864223 − 4.90125i 0.258819 + 0.965926i 0.0893866 + 2.99867i −0.855437 − 1.48166i
5.9 0.906308 + 0.422618i −0.720612 + 1.57503i 0.642788 + 0.766044i 2.23745 + 1.56668i −1.31873 + 1.12292i −0.378759 − 2.14805i 0.258819 + 0.965926i −1.96144 − 2.26997i 1.36571 + 2.36548i
5.10 0.906308 + 0.422618i 0.815069 + 1.52829i 0.642788 + 0.766044i 0.198792 + 0.139196i 0.0928210 + 1.72956i −0.0256510 − 0.145474i 0.258819 + 0.965926i −1.67133 + 2.49132i 0.121340 + 0.210167i
5.11 0.906308 + 0.422618i 1.01766 − 1.40156i 0.642788 + 0.766044i 1.16520 + 0.815881i 1.51464 − 0.840164i −0.192798 − 1.09341i 0.258819 + 0.965926i −0.928742 − 2.85262i 0.711222 + 1.23187i
5.12 0.906308 + 0.422618i 1.73164 + 0.0375568i 0.642788 + 0.766044i −1.61501 − 1.13085i 1.55353 + 0.765862i 0.209991 + 1.19092i 0.258819 + 0.965926i 2.99718 + 0.130070i −0.985784 − 1.70743i
17.1 −0.573576 + 0.819152i −1.42676 + 0.982021i −0.342020 − 0.939693i 3.24421 − 0.283831i 0.0139298 − 1.73199i 0.256080 + 0.214877i 0.965926 + 0.258819i 1.07127 − 2.80221i −1.62830 + 2.82030i
17.2 −0.573576 + 0.819152i −1.26116 − 1.18721i −0.342020 − 0.939693i −2.26391 + 0.198066i 1.69588 − 0.352130i 1.64453 + 1.37993i 0.965926 + 0.258819i 0.181071 + 2.99453i 1.13628 − 1.96809i
17.3 −0.573576 + 0.819152i −0.852694 − 1.50762i −0.342020 − 0.939693i 2.15344 − 0.188401i 1.72405 + 0.166249i −2.96483 − 2.48779i 0.965926 + 0.258819i −1.54583 + 2.57107i −1.08083 + 1.87206i
17.4 −0.573576 + 0.819152i −0.835502 + 1.51721i −0.342020 − 0.939693i −3.56866 + 0.312218i −0.763604 − 1.55464i 2.38336 + 1.99988i 0.965926 + 0.258819i −1.60387 − 2.53527i 1.79115 − 3.10236i
17.5 −0.573576 + 0.819152i 1.40979 − 1.00622i −0.342020 − 0.939693i −3.90847 + 0.341947i 0.0156231 + 1.73198i −3.88722 − 3.26176i 0.965926 + 0.258819i 0.975040 − 2.83713i 1.96170 − 3.39776i
17.6 −0.573576 + 0.819152i 1.45751 + 0.935770i −0.342020 − 0.939693i 0.419150 − 0.0366709i −1.60253 + 0.657187i 0.598452 + 0.502161i 0.965926 + 0.258819i 1.24867 + 2.72779i −0.210376 + 0.364381i
17.7 0.573576 − 0.819152i −1.59116 + 0.684268i −0.342020 − 0.939693i 2.26391 − 0.198066i −0.352130 + 1.69588i 1.64453 + 1.37993i −0.965926 − 0.258819i 2.06356 − 2.17755i 1.13628 − 1.96809i
17.8 0.573576 − 0.819152i −1.31691 + 1.12506i −0.342020 − 0.939693i −2.15344 + 0.188401i 0.166249 + 1.72405i −2.96483 − 2.48779i −0.965926 − 0.258819i 0.468481 − 2.96320i −1.08083 + 1.87206i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.i odd 36 1 inner
111.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 222.2.q.a ✓ 144
3.b odd 2 1 inner 222.2.q.a ✓ 144
37.i odd 36 1 inner 222.2.q.a ✓ 144
111.q even 36 1 inner 222.2.q.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.q.a ✓ 144 1.a even 1 1 trivial
222.2.q.a ✓ 144 3.b odd 2 1 inner
222.2.q.a ✓ 144 37.i odd 36 1 inner
222.2.q.a ✓ 144 111.q even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(222, [\chi])\).