Properties

Label 855.2.k.a
Level $855$
Weight $2$
Character orbit 855.k
Analytic conductor $6.827$
Analytic rank $0$
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: \(0\)
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 285)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 2 q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 2 \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - 2 q^{7} - 2 \zeta_{6} q^{10} - q^{11} - 2 \zeta_{6} q^{13} + ( - 4 \zeta_{6} + 4) q^{14} + ( - 4 \zeta_{6} + 4) q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + (5 \zeta_{6} - 2) q^{19} + 2 q^{20} + ( - 2 \zeta_{6} + 2) q^{22} - 4 \zeta_{6} q^{23} - \zeta_{6} q^{25} + 4 q^{26} + 4 \zeta_{6} q^{28} - 5 \zeta_{6} q^{29} + 9 q^{31} + 8 \zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + ( - 2 \zeta_{6} + 2) q^{35} - 6 q^{37} + ( - 4 \zeta_{6} - 6) q^{38} + ( - 6 \zeta_{6} + 6) q^{41} + ( - 10 \zeta_{6} + 10) q^{43} + 2 \zeta_{6} q^{44} + 8 q^{46} - 3 q^{49} + 2 q^{50} + (4 \zeta_{6} - 4) q^{52} - 2 \zeta_{6} q^{53} + ( - \zeta_{6} + 1) q^{55} + 10 q^{58} + ( - 7 \zeta_{6} + 7) q^{59} + 7 \zeta_{6} q^{61} + (18 \zeta_{6} - 18) q^{62} - 8 q^{64} + 2 q^{65} - 8 \zeta_{6} q^{67} - 4 q^{68} + 4 \zeta_{6} q^{70} + ( - 3 \zeta_{6} + 3) q^{71} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 12 \zeta_{6} + 12) q^{74} + ( - 6 \zeta_{6} + 10) q^{76} + 2 q^{77} + ( - 11 \zeta_{6} + 11) q^{79} + 4 \zeta_{6} q^{80} + 12 \zeta_{6} q^{82} - 6 q^{83} + 2 \zeta_{6} q^{85} + 20 \zeta_{6} q^{86} + 15 \zeta_{6} q^{89} + 4 \zeta_{6} q^{91} + (8 \zeta_{6} - 8) q^{92} + ( - 2 \zeta_{6} - 3) q^{95} + (8 \zeta_{6} - 8) q^{97} + ( - 6 \zeta_{6} + 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{4} - q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{4} - q^{5} - 4 q^{7} - 2 q^{10} - 2 q^{11} - 2 q^{13} + 4 q^{14} + 4 q^{16} + 2 q^{17} + q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} - q^{25} + 8 q^{26} + 4 q^{28} - 5 q^{29} + 18 q^{31} + 8 q^{32} + 4 q^{34} + 2 q^{35} - 12 q^{37} - 16 q^{38} + 6 q^{41} + 10 q^{43} + 2 q^{44} + 16 q^{46} - 6 q^{49} + 4 q^{50} - 4 q^{52} - 2 q^{53} + q^{55} + 20 q^{58} + 7 q^{59} + 7 q^{61} - 18 q^{62} - 16 q^{64} + 4 q^{65} - 8 q^{67} - 8 q^{68} + 4 q^{70} + 3 q^{71} + 2 q^{73} + 12 q^{74} + 14 q^{76} + 4 q^{77} + 11 q^{79} + 4 q^{80} + 12 q^{82} - 12 q^{83} + 2 q^{85} + 20 q^{86} + 15 q^{89} + 4 q^{91} - 8 q^{92} - 8 q^{95} - 8 q^{97} + 6 q^{98}+O(q^{100}) \) 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
−1.00000 1.73205i 0 −1.00000 + 1.73205i −0.500000 0.866025i 0 −2.00000 0 0 −1.00000 + 1.73205i
676.1 −1.00000 + 1.73205i 0 −1.00000 1.73205i −0.500000 + 0.866025i 0 −2.00000 0 0 −1.00000 1.73205i
\(n\): 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.a 2
3.b odd 2 1 285.2.i.c 2
19.c even 3 1 inner 855.2.k.a 2
57.f even 6 1 5415.2.a.l 1
57.h odd 6 1 285.2.i.c 2
57.h odd 6 1 5415.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.c 2 3.b odd 2 1
285.2.i.c 2 57.h odd 6 1
855.2.k.a 2 1.a even 1 1 trivial
855.2.k.a 2 19.c even 3 1 inner
5415.2.a.b 1 57.h odd 6 1
5415.2.a.l 1 57.f even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$31$ \( (T - 9)^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) 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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