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Properties

Label 855.2.be.f
Level $855$
Weight $2$
Character orbit 855.be
Analytic conductor $6.827$
Analytic rank $0$
Dimension $40$
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] = [855,2,Mod(64,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.be (of order \(6\), 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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 18 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 18 q^{4} + 2 q^{5} - 6 q^{10} + 16 q^{11} - 8 q^{14} - 14 q^{16} - 4 q^{19} - 36 q^{20} - 4 q^{25} - 56 q^{26} - 8 q^{29} - 64 q^{31} - 14 q^{34} + 30 q^{35} + 8 q^{40} - 12 q^{41} + 84 q^{44} - 80 q^{46} + 16 q^{50} + 4 q^{55} - 24 q^{56} - 16 q^{61} - 72 q^{64} - 4 q^{65} - 36 q^{70} + 48 q^{71} + 28 q^{74} + 76 q^{76} - 48 q^{79} - 50 q^{80} - 24 q^{85} + 64 q^{86} - 16 q^{89} + 20 q^{91} + 220 q^{94} + 34 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.32126 + 1.34018i 0 2.59217 − 4.48977i 1.71385 − 1.43622i 0 2.01196i 8.53517i 0 −2.05349 + 5.63071i
64.2 −2.22343 + 1.28370i 0 2.29575 − 3.97636i −1.71920 − 1.42980i 0 0.284179i 6.65341i 0 5.65796 + 0.972128i
64.3 −1.70934 + 0.986890i 0 0.947905 − 1.64182i −0.735691 + 2.11158i 0 − 3.30075i − 0.205648i 0 −0.826345 − 4.33546i
64.4 −1.62648 + 0.939050i 0 0.763630 − 1.32265i 1.56342 + 1.59866i 0 − 2.01092i − 0.887852i 0 −4.04410 − 1.13207i
64.5 −1.57407 + 0.908790i 0 0.651800 − 1.12895i −2.06774 + 0.851136i 0 1.60010i − 1.26576i 0 2.48127 − 3.21890i
64.6 −1.35687 + 0.783391i 0 0.227402 − 0.393873i 1.32800 − 1.79900i 0 1.30034i − 2.42098i 0 −0.392606 + 3.48136i
64.7 −0.725611 + 0.418932i 0 −0.648993 + 1.12409i 1.43263 − 1.71685i 0 4.37869i − 2.76326i 0 −0.320286 + 1.84594i
64.8 −0.709573 + 0.409672i 0 −0.664337 + 1.15067i 1.62195 + 1.53925i 0 2.60440i − 2.72733i 0 −1.78148 − 0.427742i
64.9 −0.703345 + 0.406076i 0 −0.670204 + 1.16083i −1.31948 − 1.80526i 0 − 2.73275i − 2.71292i 0 1.66112 + 0.733912i
64.10 −0.0855279 + 0.0493796i 0 −0.995123 + 1.72360i 1.97271 + 1.05281i 0 − 3.64109i − 0.394073i 0 −0.220709 + 0.00736690i
64.11 0.0855279 − 0.0493796i 0 −0.995123 + 1.72360i −0.0745946 + 2.23482i 0 3.64109i 0.394073i 0 0.103975 + 0.194823i
64.12 0.703345 − 0.406076i 0 −0.670204 + 1.16083i −0.903663 − 2.04533i 0 2.73275i 2.71292i 0 −1.46615 − 1.07162i
64.13 0.709573 − 0.409672i 0 −0.664337 + 1.15067i 0.522053 + 2.17427i 0 − 2.60440i 2.72733i 0 1.26117 + 1.32893i
64.14 0.725611 − 0.418932i 0 −0.648993 + 1.12409i −2.20315 + 0.382266i 0 − 4.37869i 2.76326i 0 −1.43849 + 1.20035i
64.15 1.35687 − 0.783391i 0 0.227402 − 0.393873i −2.22198 + 0.250581i 0 − 1.30034i 2.42098i 0 −2.81865 + 2.08069i
64.16 1.57407 − 0.908790i 0 0.651800 − 1.12895i 1.77098 − 1.36515i 0 − 1.60010i 1.26576i 0 1.54701 − 3.75829i
64.17 1.62648 − 0.939050i 0 0.763630 − 1.32265i 0.602772 + 2.15329i 0 2.01092i 0.887852i 0 3.00245 + 2.93626i
64.18 1.70934 − 0.986890i 0 0.947905 − 1.64182i 2.19653 + 0.418661i 0 3.30075i 0.205648i 0 4.16779 − 1.45209i
64.19 2.22343 − 1.28370i 0 2.29575 − 3.97636i −0.378644 − 2.20378i 0 − 0.284179i − 6.65341i 0 −3.67087 − 4.41387i
64.20 2.32126 − 1.34018i 0 2.59217 − 4.48977i −2.10073 + 0.766124i 0 − 2.01196i − 8.53517i 0 −3.84959 + 4.59373i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.be.f 40
3.b odd 2 1 285.2.r.a ✓ 40
5.b even 2 1 inner 855.2.be.f 40
15.d odd 2 1 285.2.r.a ✓ 40
19.c even 3 1 inner 855.2.be.f 40
57.h odd 6 1 285.2.r.a ✓ 40
95.i even 6 1 inner 855.2.be.f 40
285.n odd 6 1 285.2.r.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.r.a ✓ 40 3.b odd 2 1
285.2.r.a ✓ 40 15.d odd 2 1
285.2.r.a ✓ 40 57.h odd 6 1
285.2.r.a ✓ 40 285.n odd 6 1
855.2.be.f 40 1.a even 1 1 trivial
855.2.be.f 40 5.b even 2 1 inner
855.2.be.f 40 19.c even 3 1 inner
855.2.be.f 40 95.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 29 T_{2}^{38} + 491 T_{2}^{36} - 5568 T_{2}^{34} + 47175 T_{2}^{32} - 307397 T_{2}^{30} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display