Properties

Label 855.2.k.b
Level $855$
Weight $2$
Character orbit 855.k
Analytic conductor $6.827$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(406,855)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.k (of order \(3\), 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 4 q^{7} - 3 q^{11} - 2 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - 3 \zeta_{6} - 2) q^{19} - 2 q^{20} - \zeta_{6} q^{25} - 8 \zeta_{6} q^{28} - 3 \zeta_{6} q^{29} - 7 q^{31} + ( - 4 \zeta_{6} + 4) q^{35} + 8 q^{37} + (6 \zeta_{6} - 6) q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 6 \zeta_{6} q^{44} + 6 \zeta_{6} q^{47} + 9 q^{49} + ( - 4 \zeta_{6} + 4) q^{52} - 6 \zeta_{6} q^{53} + ( - 3 \zeta_{6} + 3) q^{55} + (15 \zeta_{6} - 15) q^{59} - 5 \zeta_{6} q^{61} - 8 q^{64} + 2 q^{65} - 2 \zeta_{6} q^{67} + 12 q^{68} + (3 \zeta_{6} - 3) q^{71} + (8 \zeta_{6} - 8) q^{73} + ( - 10 \zeta_{6} + 6) q^{76} + 12 q^{77} + (5 \zeta_{6} - 5) q^{79} - 4 \zeta_{6} q^{80} - 12 q^{83} + 6 \zeta_{6} q^{85} - 15 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( - 2 \zeta_{6} + 5) q^{95} + (8 \zeta_{6} - 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - q^{5} - 8 q^{7} - 6 q^{11} - 2 q^{13} - 4 q^{16} + 6 q^{17} - 7 q^{19} - 4 q^{20} - q^{25} - 8 q^{28} - 3 q^{29} - 14 q^{31} + 4 q^{35} + 16 q^{37} - 6 q^{41} + 4 q^{43} - 6 q^{44} + 6 q^{47} + 18 q^{49} + 4 q^{52} - 6 q^{53} + 3 q^{55} - 15 q^{59} - 5 q^{61} - 16 q^{64} + 4 q^{65} - 2 q^{67} + 24 q^{68} - 3 q^{71} - 8 q^{73} + 2 q^{76} + 24 q^{77} - 5 q^{79} - 4 q^{80} - 24 q^{83} + 6 q^{85} - 15 q^{89} + 8 q^{91} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 1.00000 1.73205i −0.500000 0.866025i 0 −4.00000 0 0 0
676.1 0 0 1.00000 + 1.73205i −0.500000 + 0.866025i 0 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.k.b 2
3.b odd 2 1 95.2.e.a 2
12.b even 2 1 1520.2.q.c 2
15.d odd 2 1 475.2.e.b 2
15.e even 4 2 475.2.j.a 4
19.c even 3 1 inner 855.2.k.b 2
57.f even 6 1 1805.2.a.b 1
57.h odd 6 1 95.2.e.a 2
57.h odd 6 1 1805.2.a.a 1
228.m even 6 1 1520.2.q.c 2
285.n odd 6 1 475.2.e.b 2
285.n odd 6 1 9025.2.a.g 1
285.q even 6 1 9025.2.a.e 1
285.v even 12 2 475.2.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.a 2 3.b odd 2 1
95.2.e.a 2 57.h odd 6 1
475.2.e.b 2 15.d odd 2 1
475.2.e.b 2 285.n odd 6 1
475.2.j.a 4 15.e even 4 2
475.2.j.a 4 285.v even 12 2
855.2.k.b 2 1.a even 1 1 trivial
855.2.k.b 2 19.c even 3 1 inner
1520.2.q.c 2 12.b even 2 1
1520.2.q.c 2 228.m even 6 1
1805.2.a.a 1 57.h odd 6 1
1805.2.a.b 1 57.f even 6 1
9025.2.a.e 1 285.q even 6 1
9025.2.a.g 1 285.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
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