↑

Properties

Label 1255.2.b.a
Level $1255$
Weight $2$
Character orbit 1255.b
Analytic conductor $10.021$
Analytic rank $0$
Dimension $48$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1255,2,Mod(754,1255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1255, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1255.754");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1255 = 5 \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1255.b (of order \(2\), degree \(1\), 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: \(10.0212254537\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 28 q^{4} - q^{5} - 8 q^{6} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 28 q^{4} - q^{5} - 8 q^{6} - 22 q^{9} - q^{10} - 12 q^{11} + 22 q^{14} - 2 q^{15} - 20 q^{16} + 38 q^{19} + 4 q^{20} - 54 q^{21} + 24 q^{24} + q^{25} - 28 q^{26} + 62 q^{29} + 18 q^{30} - 42 q^{31} + 26 q^{34} - q^{35} - 48 q^{36} + 56 q^{39} + 13 q^{40} - 42 q^{41} + 22 q^{44} + 13 q^{45} - 58 q^{46} + 24 q^{49} + 16 q^{50} - 64 q^{51} + 32 q^{54} - q^{55} - 48 q^{56} + 34 q^{59} + 18 q^{60} - 138 q^{61} + 78 q^{64} - q^{65} - 60 q^{66} + 64 q^{69} + 43 q^{70} - 30 q^{71} + 28 q^{74} + 23 q^{75} - 136 q^{76} + 82 q^{79} + 25 q^{80} - 48 q^{81} + 20 q^{84} + 11 q^{85} - 54 q^{86} + 44 q^{89} + 114 q^{90} - 90 q^{91} + 56 q^{94} + 32 q^{95} - 56 q^{96} + 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
754.1 − 2.56089i − 1.74331i −4.55818 −0.959361 + 2.01981i −4.46444 1.34088i 6.55123i −0.0391369 5.17252 + 2.45682i
754.2 − 2.43497i 0.451094i −3.92906 −2.23428 + 0.0893899i 1.09840 2.05475i 4.69721i 2.79651 0.217661 + 5.44040i
754.3 − 2.40622i − 2.93697i −3.78987 1.47173 − 1.68345i −7.06697 − 1.55258i 4.30682i −5.62578 −4.05074 − 3.54131i
754.4 − 2.28741i 0.741146i −3.23223 0.963065 + 2.01804i 1.69530 − 0.521650i 2.81861i 2.45070 4.61609 − 2.20292i
754.5 − 2.26964i 2.38004i −3.15128 0.547021 − 2.16813i 5.40184 4.43492i 2.61299i −2.66458 −4.92087 − 1.24154i
754.6 − 2.15117i 0.529384i −2.62752 2.19722 + 0.415017i 1.13879 2.92459i 1.34990i 2.71975 0.892770 − 4.72658i
754.7 − 2.10231i − 1.64594i −2.41970 0.338609 − 2.21028i −3.46027 − 0.327398i 0.882332i 0.290880 −4.64669 − 0.711860i
754.8 − 2.05626i 0.582973i −2.22819 −0.961060 − 2.01900i 1.19874 − 4.43029i 0.469208i 2.66014 −4.15158 + 1.97619i
754.9 − 1.82719i 1.90399i −1.33863 1.97639 + 1.04588i 3.47895 1.91569i − 1.20845i −0.625162 1.91103 − 3.61125i
754.10 − 1.63195i − 1.67948i −0.663261 −1.60704 − 1.55481i −2.74083 − 1.08318i − 2.18149i 0.179335 −2.53737 + 2.62261i
754.11 − 1.49173i − 0.351479i −0.225245 −1.88584 + 1.20150i −0.524310 − 3.90188i − 2.64745i 2.87646 1.79230 + 2.81316i
754.12 − 1.43197i − 2.11641i −0.0505501 0.782366 + 2.09473i −3.03065 3.18647i − 2.79156i −1.47919 2.99960 − 1.12033i
754.13 − 1.40234i − 1.11395i 0.0334481 −1.95799 + 1.07995i −1.56213 2.69191i − 2.85158i 1.75912 1.51445 + 2.74576i
754.14 − 1.40070i 2.17144i 0.0380356 −0.785069 + 2.09372i 3.04154 − 1.93816i − 2.85468i −1.71516 2.93268 + 1.09965i
754.15 − 1.33572i 2.04729i 0.215852 −2.02501 − 0.948333i 2.73461 0.616134i − 2.95976i −1.19141 −1.26671 + 2.70485i
754.16 − 1.01731i 2.41641i 0.965071 2.14229 − 0.640780i 2.45825 − 0.622404i − 3.01641i −2.83902 −0.651875 − 2.17938i
754.17 − 0.953949i 0.0478012i 1.08998 1.42888 + 1.71997i 0.0455999 − 1.87913i − 2.94769i 2.99772 1.64076 − 1.36308i
754.18 − 0.888830i − 3.07161i 1.20998 −0.753646 − 2.10524i −2.73013 0.475305i − 2.85313i −6.43476 −1.87120 + 0.669863i
754.19 − 0.887122i − 1.83396i 1.21301 −1.81353 − 1.30810i −1.62695 − 1.97145i − 2.85034i −0.363418 −1.16044 + 1.60882i
754.20 − 0.580765i − 0.394595i 1.66271 1.99445 + 1.01102i −0.229167 2.25537i − 2.12718i 2.84429 0.587163 − 1.15831i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 754.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1255.2.b.a ✓ 48
5.b even 2 1 inner 1255.2.b.a ✓ 48
5.c odd 4 2 6275.2.a.m 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1255.2.b.a ✓ 48 1.a even 1 1 trivial
1255.2.b.a ✓ 48 5.b even 2 1 inner
6275.2.a.m 48 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 62 T_{2}^{46} + 1789 T_{2}^{44} + 31917 T_{2}^{42} + 394512 T_{2}^{40} + 3588111 T_{2}^{38} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(1255, [\chi])\). Copy content Toggle raw display