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Properties

Label 255.2.w.b
Level $255$
Weight $2$
Character orbit 255.w
Analytic conductor $2.036$
Analytic rank $0$
Dimension $32$
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] = [255,2,Mod(76,255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(255, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("255.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 255 = 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 255.w (of order \(8\), degree \(4\), 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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} + 16 q^{12} - 56 q^{16} + 8 q^{17} + 8 q^{18} - 16 q^{19} + 8 q^{22} - 72 q^{26} - 24 q^{28} + 32 q^{29} - 48 q^{33} + 8 q^{34} - 16 q^{37} - 8 q^{39} - 24 q^{40} + 32 q^{41} - 8 q^{42} - 16 q^{43} + 16 q^{44} + 8 q^{46} + 8 q^{49} - 8 q^{50} + 8 q^{51} + 128 q^{52} - 8 q^{53} + 16 q^{56} + 112 q^{58} - 24 q^{59} - 48 q^{61} - 8 q^{62} - 8 q^{63} + 8 q^{66} - 48 q^{67} + 96 q^{68} + 32 q^{69} + 8 q^{70} - 32 q^{71} - 32 q^{73} - 88 q^{74} + 24 q^{76} - 88 q^{77} + 16 q^{78} + 24 q^{79} + 64 q^{80} + 8 q^{82} + 24 q^{83} + 24 q^{84} + 8 q^{85} + 48 q^{86} - 8 q^{87} - 112 q^{88} + 80 q^{91} + 152 q^{92} + 32 q^{93} - 104 q^{94} + 48 q^{96} - 24 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −1.91928 + 1.91928i −0.382683 + 0.923880i − 5.36729i −0.923880 − 0.382683i −1.03871 − 2.50766i 3.53907 − 1.46593i 6.46277 + 6.46277i −0.707107 − 0.707107i 2.50766 − 1.03871i
76.2 −1.25280 + 1.25280i 0.382683 − 0.923880i − 1.13900i 0.923880 + 0.382683i 0.678009 + 1.63686i 3.81159 − 1.57881i −1.07866 − 1.07866i −0.707107 − 0.707107i −1.63686 + 0.678009i
76.3 −1.07263 + 1.07263i −0.382683 + 0.923880i − 0.301074i −0.923880 − 0.382683i −0.580504 − 1.40146i −1.95554 + 0.810011i −1.82232 − 1.82232i −0.707107 − 0.707107i 1.40146 − 0.580504i
76.4 −1.06986 + 1.06986i 0.382683 − 0.923880i − 0.289222i 0.923880 + 0.382683i 0.579007 + 1.39785i −4.38000 + 1.81426i −1.83030 − 1.83030i −0.707107 − 0.707107i −1.39785 + 0.579007i
76.5 0.245139 − 0.245139i 0.382683 − 0.923880i 1.87981i 0.923880 + 0.382683i −0.132669 − 0.320290i 0.860352 − 0.356370i 0.951095 + 0.951095i −0.707107 − 0.707107i 0.320290 − 0.132669i
76.6 0.470313 − 0.470313i −0.382683 + 0.923880i 1.55761i −0.923880 − 0.382683i 0.254531 + 0.614493i −1.36091 + 0.563709i 1.67319 + 1.67319i −0.707107 − 0.707107i −0.614493 + 0.254531i
76.7 1.37042 − 1.37042i 0.382683 − 0.923880i − 1.75608i 0.923880 + 0.382683i −0.741663 − 1.79053i 0.108602 − 0.0449843i 0.334278 + 0.334278i −0.707107 − 0.707107i 1.79053 − 0.741663i
76.8 1.81449 − 1.81449i −0.382683 + 0.923880i − 4.58477i −0.923880 − 0.382683i 0.981997 + 2.37075i 2.79106 − 1.15609i −4.69005 − 4.69005i −0.707107 − 0.707107i −2.37075 + 0.981997i
121.1 −1.86360 − 1.86360i 0.923880 − 0.382683i 4.94602i −0.382683 − 0.923880i −2.43491 − 1.00857i −0.0311391 + 0.0751765i 5.49020 − 5.49020i 0.707107 − 0.707107i −1.00857 + 2.43491i
121.2 −1.37952 − 1.37952i −0.923880 + 0.382683i 1.80613i 0.382683 + 0.923880i 1.80243 + 0.746589i 1.12408 − 2.71378i −0.267441 + 0.267441i 0.707107 − 0.707107i 0.746589 − 1.80243i
121.3 −0.392111 − 0.392111i −0.923880 + 0.382683i − 1.69250i 0.382683 + 0.923880i 0.512318 + 0.212209i −2.00796 + 4.84764i −1.44787 + 1.44787i 0.707107 − 0.707107i 0.212209 − 0.512318i
121.4 −0.0443119 − 0.0443119i 0.923880 − 0.382683i − 1.99607i −0.382683 − 0.923880i −0.0578963 − 0.0239814i −0.295776 + 0.714067i −0.177073 + 0.177073i 0.707107 − 0.707107i −0.0239814 + 0.0578963i
121.5 0.689027 + 0.689027i −0.923880 + 0.382683i − 1.05048i 0.382683 + 0.923880i −0.900257 − 0.372899i 1.50637 − 3.63669i 2.10187 − 2.10187i 0.707107 − 0.707107i −0.372899 + 0.900257i
121.6 0.876678 + 0.876678i 0.923880 − 0.382683i − 0.462871i −0.382683 − 0.923880i 1.14544 + 0.474455i 0.999316 − 2.41256i 2.15915 − 2.15915i 0.707107 − 0.707107i 0.474455 − 1.14544i
121.7 1.73834 + 1.73834i 0.923880 − 0.382683i 4.04366i −0.382683 − 0.923880i 2.27125 + 0.940784i −0.920704 + 2.22278i −3.55258 + 3.55258i 0.707107 − 0.707107i 0.940784 − 2.27125i
121.8 1.78971 + 1.78971i −0.923880 + 0.382683i 4.40611i 0.382683 + 0.923880i −2.33837 − 0.968583i 0.211600 − 0.510848i −4.30624 + 4.30624i 0.707107 − 0.707107i −0.968583 + 2.33837i
151.1 −1.91928 − 1.91928i −0.382683 − 0.923880i 5.36729i −0.923880 + 0.382683i −1.03871 + 2.50766i 3.53907 + 1.46593i 6.46277 − 6.46277i −0.707107 + 0.707107i 2.50766 + 1.03871i
151.2 −1.25280 − 1.25280i 0.382683 + 0.923880i 1.13900i 0.923880 − 0.382683i 0.678009 − 1.63686i 3.81159 + 1.57881i −1.07866 + 1.07866i −0.707107 + 0.707107i −1.63686 − 0.678009i
151.3 −1.07263 − 1.07263i −0.382683 − 0.923880i 0.301074i −0.923880 + 0.382683i −0.580504 + 1.40146i −1.95554 − 0.810011i −1.82232 + 1.82232i −0.707107 + 0.707107i 1.40146 + 0.580504i
151.4 −1.06986 − 1.06986i 0.382683 + 0.923880i 0.289222i 0.923880 − 0.382683i 0.579007 − 1.39785i −4.38000 − 1.81426i −1.83030 + 1.83030i −0.707107 + 0.707107i −1.39785 − 0.579007i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 255.2.w.b ✓ 32
3.b odd 2 1 765.2.be.c 32
17.d even 8 1 inner 255.2.w.b ✓ 32
17.e odd 16 1 4335.2.a.bl 16
17.e odd 16 1 4335.2.a.bm 16
51.g odd 8 1 765.2.be.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.w.b ✓ 32 1.a even 1 1 trivial
255.2.w.b ✓ 32 17.d even 8 1 inner
765.2.be.c 32 3.b odd 2 1
765.2.be.c 32 51.g odd 8 1
4335.2.a.bl 16 17.e odd 16 1
4335.2.a.bm 16 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 138 T_{2}^{28} + 24 T_{2}^{25} + 6837 T_{2}^{24} + 320 T_{2}^{23} + 296 T_{2}^{21} + \cdots + 289 \) acting on \(S_{2}^{\mathrm{new}}(255, [\chi])\). Copy content Toggle raw display