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Properties

Label 816.2.cg.a
Level $816$
Weight $2$
Character orbit 816.cg
Analytic conductor $6.516$
Analytic rank $0$
Dimension $576$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(91,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 4, 0, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.cg (of order \(16\), degree \(8\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(576\)
Relative dimension: \(72\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q + 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q + 16 q^{6} - 32 q^{19} - 16 q^{24} + 48 q^{28} + 64 q^{30} + 16 q^{36} - 80 q^{38} - 176 q^{40} + 64 q^{44} + 64 q^{58} - 16 q^{60} + 64 q^{61} - 80 q^{62} - 96 q^{64} + 64 q^{65} + 64 q^{70} - 96 q^{82} + 96 q^{84} + 160 q^{92} + 112 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −1.41385 + 0.0321328i −0.831470 − 0.555570i 1.99793 − 0.0908618i −0.563041 + 0.111996i 1.19342 + 0.758775i −0.308310 − 0.461419i −2.82186 + 0.192664i 0.382683 + 0.923880i 0.792456 − 0.176437i
91.2 −1.41336 − 0.0491415i −0.831470 − 0.555570i 1.99517 + 0.138909i 3.32256 − 0.660898i 1.14786 + 0.826080i 0.258750 + 0.387246i −2.81307 − 0.294374i 0.382683 + 0.923880i −4.72845 + 0.770811i
91.3 −1.40775 + 0.135079i 0.831470 + 0.555570i 1.96351 − 0.380315i −3.55259 + 0.706655i −1.24555 − 0.669788i 2.58022 + 3.86158i −2.71275 + 0.800617i 0.382683 + 0.923880i 4.90570 − 1.47467i
91.4 −1.40068 + 0.195166i 0.831470 + 0.555570i 1.92382 − 0.546730i 0.707547 − 0.140740i −1.27305 − 0.615903i 0.942115 + 1.40998i −2.58796 + 1.14126i 0.382683 + 0.923880i −0.963581 + 0.335221i
91.5 −1.37078 + 0.347794i 0.831470 + 0.555570i 1.75808 − 0.953499i 3.38398 − 0.673116i −1.33299 − 0.472385i −1.75325 − 2.62392i −2.07832 + 1.91849i 0.382683 + 0.923880i −4.40459 + 2.09962i
91.6 −1.35545 − 0.403432i 0.831470 + 0.555570i 1.67448 + 1.09366i −1.70423 + 0.338992i −0.902880 − 1.08849i −0.635751 − 0.951469i −1.82846 − 2.15795i 0.382683 + 0.923880i 2.44675 + 0.228054i
91.7 −1.34815 − 0.427176i −0.831470 − 0.555570i 1.63504 + 1.15180i 1.32006 − 0.262576i 0.883624 + 1.10418i 1.71446 + 2.56587i −1.71227 − 2.25125i 0.382683 + 0.923880i −1.89181 − 0.209904i
91.8 −1.34630 − 0.432991i 0.831470 + 0.555570i 1.62504 + 1.16587i 0.796334 − 0.158401i −0.878849 − 1.10798i −2.05842 − 3.08064i −1.68297 − 2.27324i 0.382683 + 0.923880i −1.14069 − 0.131551i
91.9 −1.32905 + 0.483345i −0.831470 − 0.555570i 1.53275 − 1.28478i −2.60235 + 0.517640i 1.37360 + 0.336495i 1.14941 + 1.72021i −1.41612 + 2.44839i 0.382683 + 0.923880i 3.20846 − 1.94581i
91.10 −1.27428 + 0.613370i 0.831470 + 0.555570i 1.24756 − 1.56320i −0.544322 + 0.108272i −1.40029 − 0.197951i −0.0321072 − 0.0480519i −0.630906 + 2.75716i 0.382683 + 0.923880i 0.627205 − 0.471840i
91.11 −1.25022 + 0.661030i −0.831470 − 0.555570i 1.12608 − 1.65286i −1.02846 + 0.204572i 1.40676 + 0.144956i −1.32044 − 1.97618i −0.315251 + 2.81080i 0.382683 + 0.923880i 1.15056 − 0.935599i
91.12 −1.24573 + 0.669447i 0.831470 + 0.555570i 1.10368 − 1.66790i −3.97419 + 0.790515i −1.40771 − 0.135466i −2.38288 − 3.56624i −0.258321 + 2.81661i 0.382683 + 0.923880i 4.42155 − 3.64527i
91.13 −1.22448 − 0.707574i −0.831470 − 0.555570i 0.998679 + 1.73281i −0.323872 + 0.0644222i 0.625007 + 1.26861i −1.39608 − 2.08938i 0.00323564 − 2.82843i 0.382683 + 0.923880i 0.442157 + 0.150280i
91.14 −1.19197 − 0.761056i −0.831470 − 0.555570i 0.841588 + 1.81431i −4.03288 + 0.802189i 0.568267 + 1.29502i −2.05612 − 3.07720i 0.377646 − 2.80310i 0.382683 + 0.923880i 5.41758 + 2.11306i
91.15 −1.19078 + 0.762917i −0.831470 − 0.555570i 0.835914 − 1.81693i 3.90606 − 0.776964i 1.41395 + 0.0272193i −1.95186 − 2.92117i 0.390780 + 2.80130i 0.382683 + 0.923880i −4.05850 + 3.90520i
91.16 −1.10832 + 0.878421i 0.831470 + 0.555570i 0.456754 − 1.94715i 2.92053 − 0.580929i −1.40956 + 0.114630i 1.92267 + 2.87748i 1.20418 + 2.55929i 0.382683 + 0.923880i −2.72658 + 3.20931i
91.17 −1.04888 − 0.948603i 0.831470 + 0.555570i 0.200306 + 1.98994i 1.71746 − 0.341623i −0.345098 − 1.37146i 1.48782 + 2.22667i 1.67757 − 2.27723i 0.382683 + 0.923880i −2.12547 − 1.27086i
91.18 −0.995977 − 1.00401i −0.831470 − 0.555570i −0.0160581 + 1.99994i 3.77446 − 0.750786i 0.270329 + 1.38814i −0.720022 − 1.07759i 2.02394 − 1.97577i 0.382683 + 0.923880i −4.51307 − 3.04181i
91.19 −0.965795 − 1.03307i −0.831470 − 0.555570i −0.134481 + 1.99547i −1.62106 + 0.322449i 0.229084 + 1.39554i 2.13530 + 3.19571i 2.19135 − 1.78829i 0.382683 + 0.923880i 1.89873 + 1.36326i
91.20 −0.958232 + 1.04009i 0.831470 + 0.555570i −0.163583 − 1.99330i −1.72980 + 0.344079i −1.37458 + 0.332440i 0.160326 + 0.239945i 2.22996 + 1.73990i 0.382683 + 0.923880i 1.29968 − 2.12886i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
272.bd even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.cg.a ✓ 576
16.f odd 4 1 816.2.cs.a yes 576
17.e odd 16 1 816.2.cs.a yes 576
272.bd even 16 1 inner 816.2.cg.a ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.cg.a ✓ 576 1.a even 1 1 trivial
816.2.cg.a ✓ 576 272.bd even 16 1 inner
816.2.cs.a yes 576 16.f odd 4 1
816.2.cs.a yes 576 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(816, [\chi])\).