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Properties

Label 255.2.r.b
Level $255$
Weight $2$
Character orbit 255.r
Analytic conductor $2.036$
Analytic rank $0$
Dimension $56$
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] = [255,2,Mod(38,255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(255, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("255.38");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 255 = 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 255.r (of order \(4\), degree \(2\), 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: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{3} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{3} + 16 q^{9} - 24 q^{10} - 12 q^{13} + 20 q^{15} - 76 q^{16} - 4 q^{18} - 16 q^{21} + 28 q^{24} - 36 q^{25} - 80 q^{27} + 20 q^{30} - 40 q^{31} + 12 q^{33} - 48 q^{34} - 24 q^{39} + 112 q^{40} - 36 q^{42} + 36 q^{43} + 8 q^{45} + 56 q^{46} + 48 q^{48} + 64 q^{49} + 8 q^{51} + 12 q^{52} + 36 q^{54} - 8 q^{55} + 16 q^{57} + 28 q^{60} - 24 q^{61} - 116 q^{63} + 16 q^{67} - 20 q^{70} + 8 q^{72} - 8 q^{73} - 12 q^{75} + 64 q^{79} + 40 q^{81} - 44 q^{85} + 72 q^{88} + 8 q^{90} + 32 q^{91} + 52 q^{93} - 244 q^{94} + 16 q^{96} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −1.91336 + 1.91336i −1.72901 − 0.102528i − 5.32191i 2.20117 − 0.393483i 3.50440 − 3.11206i −2.25039 6.35603 + 6.35603i 2.97898 + 0.354544i −3.45877 + 4.96452i
38.2 −1.78648 + 1.78648i 0.650670 − 1.60519i − 4.38303i −0.325332 + 2.21227i 1.70523 + 4.03005i −0.449944 4.25724 + 4.25724i −2.15326 − 2.08889i −3.37099 − 4.53339i
38.3 −1.72861 + 1.72861i 1.69645 − 0.349360i − 3.97617i −0.645777 − 2.14079i −2.32859 + 3.53641i −3.60614 3.41603 + 3.41603i 2.75590 − 1.18534i 4.81688 + 2.58429i
38.4 −1.71250 + 1.71250i 0.524916 + 1.65059i − 3.86531i 1.37878 − 1.76039i −3.72556 − 1.92773i 3.87611 3.19434 + 3.19434i −2.44893 + 1.73285i 0.653493 + 5.37583i
38.5 −1.27558 + 1.27558i 1.27466 + 1.17271i − 1.25418i 1.56075 + 1.60127i −3.12180 + 0.130048i −3.10696 −0.951344 − 0.951344i 0.249516 + 2.98961i −4.03339 − 0.0516953i
38.6 −1.25631 + 1.25631i −1.40831 − 1.00829i − 1.15662i −0.455929 + 2.18909i 3.03600 − 0.502553i 1.83113 −1.05954 − 1.05954i 0.966699 + 2.83998i −2.17739 − 3.32296i
38.7 −1.09029 + 1.09029i 1.49690 − 0.871367i − 0.377476i 2.16899 + 0.543598i −0.682017 + 2.58211i 2.77933 −1.76903 − 1.76903i 1.48144 − 2.60870i −2.95751 + 1.77215i
38.8 −1.01999 + 1.01999i −1.70687 + 0.294256i − 0.0807529i −0.930150 − 2.03343i 1.44085 − 2.04113i 1.62292 −1.95761 − 1.95761i 2.82683 − 1.00451i 3.02281 + 1.12533i
38.9 −0.727065 + 0.727065i 0.779167 + 1.54690i 0.942754i −1.94229 − 1.10792i −1.69120 − 0.558192i −1.16066 −2.13957 − 2.13957i −1.78580 + 2.41059i 2.21771 − 0.606642i
38.10 −0.650943 + 0.650943i 0.348580 − 1.69661i 1.15255i −2.22389 + 0.233091i 0.877492 + 1.33130i −1.96309 −2.05213 − 2.05213i −2.75698 − 1.18281i 1.29589 − 1.59935i
38.11 −0.359068 + 0.359068i −1.55046 + 0.772055i 1.74214i 0.545571 + 2.16849i 0.279501 − 0.833942i −4.75030 −1.34368 − 1.34368i 1.80786 − 2.39408i −0.974533 − 0.582739i
38.12 −0.358332 + 0.358332i 0.223832 + 1.71753i 1.74320i −0.600743 + 2.15386i −0.695650 − 0.535238i 4.23395 −1.34131 − 1.34131i −2.89980 + 0.768875i −0.556530 − 0.987061i
38.13 −0.282941 + 0.282941i −1.28501 + 1.16136i 1.83989i 2.21423 − 0.311739i 0.0349875 − 0.692178i 3.38433 −1.08646 − 1.08646i 0.302509 − 2.98471i −0.538294 + 0.714701i
38.14 −0.0502900 + 0.0502900i 1.68449 + 0.403086i 1.99494i 0.859014 − 2.06448i −0.104984 + 0.0644420i −0.440273 −0.200906 − 0.200906i 2.67504 + 1.35799i 0.0606231 + 0.147023i
38.15 0.0502900 − 0.0502900i 1.68449 − 0.403086i 1.99494i −0.859014 + 2.06448i 0.0644420 − 0.104984i −0.440273 0.200906 + 0.200906i 2.67504 − 1.35799i 0.0606231 + 0.147023i
38.16 0.282941 − 0.282941i −1.28501 − 1.16136i 1.83989i −2.21423 + 0.311739i −0.692178 + 0.0349875i 3.38433 1.08646 + 1.08646i 0.302509 + 2.98471i −0.538294 + 0.714701i
38.17 0.358332 − 0.358332i 0.223832 − 1.71753i 1.74320i 0.600743 − 2.15386i −0.535238 − 0.695650i 4.23395 1.34131 + 1.34131i −2.89980 − 0.768875i −0.556530 − 0.987061i
38.18 0.359068 − 0.359068i −1.55046 − 0.772055i 1.74214i −0.545571 − 2.16849i −0.833942 + 0.279501i −4.75030 1.34368 + 1.34368i 1.80786 + 2.39408i −0.974533 − 0.582739i
38.19 0.650943 − 0.650943i 0.348580 + 1.69661i 1.15255i 2.22389 − 0.233091i 1.33130 + 0.877492i −1.96309 2.05213 + 2.05213i −2.75698 + 1.18281i 1.29589 − 1.59935i
38.20 0.727065 − 0.727065i 0.779167 − 1.54690i 0.942754i 1.94229 + 1.10792i −0.558192 − 1.69120i −1.16066 2.13957 + 2.13957i −1.78580 − 2.41059i 2.21771 − 0.606642i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
85.i odd 4 1 inner
255.r even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 255.2.r.b yes 56
3.b odd 2 1 inner 255.2.r.b yes 56
5.c odd 4 1 255.2.k.b ✓ 56
15.e even 4 1 255.2.k.b ✓ 56
17.c even 4 1 255.2.k.b ✓ 56
51.f odd 4 1 255.2.k.b ✓ 56
85.i odd 4 1 inner 255.2.r.b yes 56
255.r even 4 1 inner 255.2.r.b yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.k.b ✓ 56 5.c odd 4 1
255.2.k.b ✓ 56 15.e even 4 1
255.2.k.b ✓ 56 17.c even 4 1
255.2.k.b ✓ 56 51.f odd 4 1
255.2.r.b yes 56 1.a even 1 1 trivial
255.2.r.b yes 56 3.b odd 2 1 inner
255.2.r.b yes 56 85.i odd 4 1 inner
255.2.r.b yes 56 255.r even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 197 T_{2}^{52} + 15776 T_{2}^{48} + 662889 T_{2}^{44} + 15797215 T_{2}^{40} + 217817082 T_{2}^{36} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(255, [\chi])\). Copy content Toggle raw display