↑

Properties

Label 255.2.bi.a
Level $255$
Weight $2$
Character orbit 255.bi
Analytic conductor $2.036$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [255,2,Mod(22,255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(255, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("255.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 255 = 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 255.bi (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: \(2.03618525154\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 16 q^{10} + 64 q^{14} - 32 q^{19} - 32 q^{20} - 16 q^{25} - 64 q^{26} - 64 q^{28} - 64 q^{30} - 32 q^{31} + 16 q^{33} + 32 q^{34} - 16 q^{36} - 32 q^{37} + 112 q^{40} - 80 q^{41} - 16 q^{46} - 64 q^{50} - 64 q^{52} + 48 q^{53} + 16 q^{54} - 64 q^{55} - 32 q^{57} - 112 q^{58} + 32 q^{59} - 32 q^{64} + 64 q^{67} - 272 q^{68} - 32 q^{70} + 32 q^{71} + 48 q^{72} - 80 q^{73} + 80 q^{74} + 32 q^{75} + 64 q^{76} - 96 q^{77} + 64 q^{78} + 32 q^{79} + 336 q^{80} - 16 q^{83} - 96 q^{84} + 32 q^{85} + 64 q^{86} + 48 q^{87} - 80 q^{88} + 320 q^{92} + 128 q^{93} + 16 q^{94} - 64 q^{95} + 64 q^{96} - 16 q^{97} + 144 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −1.06692 + 2.57576i 0.980785 + 0.195090i −4.08204 − 4.08204i 1.60405 − 1.55789i −1.54892 + 2.31813i 1.61390 − 2.41537i 9.71803 − 4.02534i 0.923880 + 0.382683i 2.30136 + 5.79380i
22.2 −0.953583 + 2.30215i −0.980785 − 0.195090i −2.97637 − 2.97637i −1.95029 + 1.09379i 1.38439 − 2.07188i 2.00084 − 2.99447i 5.08598 − 2.10668i 0.923880 + 0.382683i −0.658306 − 5.53288i
22.3 −0.907200 + 2.19017i −0.980785 − 0.195090i −2.55964 − 2.55964i 1.80249 + 1.32326i 1.31705 − 1.97110i −2.25768 + 3.37886i 3.54781 − 1.46955i 0.923880 + 0.382683i −4.53340 + 2.74730i
22.4 −0.778766 + 1.88011i 0.980785 + 0.195090i −1.51411 − 1.51411i −0.159674 + 2.23036i −1.13059 + 1.69205i −0.205092 + 0.306942i 0.265620 − 0.110024i 0.923880 + 0.382683i −4.06897 − 2.03713i
22.5 −0.773361 + 1.86706i −0.980785 − 0.195090i −1.47361 − 1.47361i 0.367583 − 2.20565i 1.12275 − 1.68031i 0.743645 − 1.11294i 0.156830 − 0.0649613i 0.923880 + 0.382683i 3.83380 + 2.39206i
22.6 −0.370807 + 0.895206i 0.980785 + 0.195090i 0.750317 + 0.750317i 0.233202 − 2.22387i −0.538328 + 0.805664i −2.65405 + 3.97206i −2.74032 + 1.13508i 0.923880 + 0.382683i 1.90435 + 1.03339i
22.7 −0.327380 + 0.790366i −0.980785 − 0.195090i 0.896713 + 0.896713i −2.07408 − 0.835580i 0.475283 − 0.711311i −1.12055 + 1.67703i −2.58303 + 1.06993i 0.923880 + 0.382683i 1.33943 − 1.36573i
22.8 −0.302018 + 0.729136i −0.980785 − 0.195090i 0.973789 + 0.973789i 1.78448 + 1.34745i 0.438462 − 0.656205i 2.14667 − 3.21272i −2.46240 + 1.01996i 0.923880 + 0.382683i −1.52142 + 0.894177i
22.9 −0.149840 + 0.361745i 0.980785 + 0.195090i 1.30581 + 1.30581i −0.280685 − 2.21838i −0.217534 + 0.325562i 2.57772 − 3.85783i −1.39152 + 0.576387i 0.923880 + 0.382683i 0.844546 + 0.230865i
22.10 −0.0361477 + 0.0872683i 0.980785 + 0.195090i 1.40790 + 1.40790i 1.81409 + 1.30731i −0.0524783 + 0.0785394i −0.273070 + 0.408678i −0.348295 + 0.144268i 0.923880 + 0.382683i −0.179662 + 0.111056i
22.11 0.163334 − 0.394324i 0.980785 + 0.195090i 1.28540 + 1.28540i −1.92546 + 1.13693i 0.237124 − 0.354882i −0.223625 + 0.334679i 1.50546 − 0.623582i 0.923880 + 0.382683i 0.133825 + 0.944953i
22.12 0.257843 − 0.622489i −0.980785 − 0.195090i 1.09320 + 1.09320i −0.0217448 − 2.23596i −0.374330 + 0.560225i 0.526551 − 0.788039i 2.20736 − 0.914319i 0.923880 + 0.382683i −1.39747 − 0.562992i
22.13 0.415831 − 1.00391i −0.980785 − 0.195090i 0.579303 + 0.579303i −0.801709 + 2.08741i −0.603693 + 0.903491i −1.60873 + 2.40764i 2.83027 − 1.17234i 0.923880 + 0.382683i 1.76218 + 1.67285i
22.14 0.606215 − 1.46353i −0.980785 − 0.195090i −0.360215 − 0.360215i 1.88548 + 1.20206i −0.880087 + 1.31714i 0.340371 − 0.509402i 2.18151 − 0.903611i 0.923880 + 0.382683i 2.90226 − 2.03076i
22.15 0.712210 − 1.71943i 0.980785 + 0.195090i −1.03497 − 1.03497i 1.99907 − 1.00186i 1.03397 − 1.54744i −1.56325 + 2.33957i 0.922181 − 0.381980i 0.923880 + 0.382683i −0.298874 − 4.15079i
22.16 0.747379 − 1.80433i 0.980785 + 0.195090i −1.28283 − 1.28283i −0.326193 + 2.21215i 1.08503 − 1.62386i 1.72927 − 2.58803i 0.335265 − 0.138871i 0.923880 + 0.382683i 3.74766 + 2.24187i
22.17 0.964591 − 2.32873i −0.980785 − 0.195090i −3.07833 − 3.07833i 1.41851 − 1.72854i −1.40037 + 2.09580i 0.101907 − 0.152514i −5.48046 + 2.27008i 0.923880 + 0.382683i −2.65701 − 4.97065i
22.18 1.03325 − 2.49449i 0.980785 + 0.195090i −3.74064 − 3.74064i −1.65882 − 1.49944i 1.50005 − 2.24498i −0.128784 + 0.192738i −8.20701 + 3.39946i 0.923880 + 0.382683i −5.45431 + 2.58860i
28.1 −2.43738 + 1.00960i 0.555570 − 0.831470i 3.50733 − 3.50733i −2.12055 − 0.709402i −0.514688 + 2.58751i −0.612966 + 3.08159i −2.98852 + 7.21492i −0.382683 − 0.923880i 5.88481 − 0.411820i
28.2 −2.32342 + 0.962393i −0.555570 + 0.831470i 3.05788 − 3.05788i 0.793770 + 2.09044i 0.490624 − 2.46653i −0.0313301 + 0.157507i −2.23707 + 5.40077i −0.382683 − 0.923880i −3.85609 − 4.09305i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.r even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 255.2.bi.a yes 144
3.b odd 2 1 765.2.cc.b 144
5.c odd 4 1 255.2.bd.a ✓ 144
15.e even 4 1 765.2.bx.b 144
17.e odd 16 1 255.2.bd.a ✓ 144
51.i even 16 1 765.2.bx.b 144
85.r even 16 1 inner 255.2.bi.a yes 144
255.bj odd 16 1 765.2.cc.b 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.bd.a ✓ 144 5.c odd 4 1
255.2.bd.a ✓ 144 17.e odd 16 1
255.2.bi.a yes 144 1.a even 1 1 trivial
255.2.bi.a yes 144 85.r even 16 1 inner
765.2.bx.b 144 15.e even 4 1
765.2.bx.b 144 51.i even 16 1
765.2.cc.b 144 3.b odd 2 1
765.2.cc.b 144 255.bj odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(255, [\chi])\).