Properties

Label 429.2.i.b
Level $429$
Weight $2$
Character orbit 429.i
Analytic conductor $3.426$
Analytic rank $0$
Dimension $2$
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] = [429,2,Mod(100,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.i (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: \(3.42558224671\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} + 3 \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{4} + 2 q^{5} + 3 \zeta_{6} q^{7} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{11} + 2 q^{12} + (3 \zeta_{6} - 4) q^{13} + ( - 2 \zeta_{6} + 2) q^{15} + (4 \zeta_{6} - 4) q^{16} - 2 \zeta_{6} q^{17} - 4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{20} + 3 q^{21} + ( - 4 \zeta_{6} + 4) q^{23} - q^{25} - q^{27} + (6 \zeta_{6} - 6) q^{28} + 5 q^{31} + \zeta_{6} q^{33} + 6 \zeta_{6} q^{35} + ( - 2 \zeta_{6} + 2) q^{36} + ( - 6 \zeta_{6} + 6) q^{37} + (4 \zeta_{6} - 1) q^{39} + ( - 8 \zeta_{6} + 8) q^{41} - 5 \zeta_{6} q^{43} - 2 q^{44} - 2 \zeta_{6} q^{45} + 4 \zeta_{6} q^{48} + (2 \zeta_{6} - 2) q^{49} - 2 q^{51} + ( - 2 \zeta_{6} - 6) q^{52} + 14 q^{53} + (2 \zeta_{6} - 2) q^{55} - 4 q^{57} + 4 q^{60} + 7 \zeta_{6} q^{61} + ( - 3 \zeta_{6} + 3) q^{63} - 8 q^{64} + (6 \zeta_{6} - 8) q^{65} + (3 \zeta_{6} - 3) q^{67} + ( - 4 \zeta_{6} + 4) q^{68} - 4 \zeta_{6} q^{69} - 10 \zeta_{6} q^{71} - 9 q^{73} + (\zeta_{6} - 1) q^{75} + ( - 8 \zeta_{6} + 8) q^{76} - 3 q^{77} - 11 q^{79} + (8 \zeta_{6} - 8) q^{80} + (\zeta_{6} - 1) q^{81} - 6 q^{83} + 6 \zeta_{6} q^{84} - 4 \zeta_{6} q^{85} + ( - 8 \zeta_{6} + 8) q^{89} + ( - 3 \zeta_{6} - 9) q^{91} + 8 q^{92} + ( - 5 \zeta_{6} + 5) q^{93} - 8 \zeta_{6} q^{95} + 7 \zeta_{6} q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{7} - q^{9} - q^{11} + 4 q^{12} - 5 q^{13} + 2 q^{15} - 4 q^{16} - 2 q^{17} - 4 q^{19} + 4 q^{20} + 6 q^{21} + 4 q^{23} - 2 q^{25} - 2 q^{27} - 6 q^{28} + 10 q^{31} + q^{33} + 6 q^{35} + 2 q^{36} + 6 q^{37} + 2 q^{39} + 8 q^{41} - 5 q^{43} - 4 q^{44} - 2 q^{45} + 4 q^{48} - 2 q^{49} - 4 q^{51} - 14 q^{52} + 28 q^{53} - 2 q^{55} - 8 q^{57} + 8 q^{60} + 7 q^{61} + 3 q^{63} - 16 q^{64} - 10 q^{65} - 3 q^{67} + 4 q^{68} - 4 q^{69} - 10 q^{71} - 18 q^{73} - q^{75} + 8 q^{76} - 6 q^{77} - 22 q^{79} - 8 q^{80} - q^{81} - 12 q^{83} + 6 q^{84} - 4 q^{85} + 8 q^{89} - 21 q^{91} + 16 q^{92} + 5 q^{93} - 8 q^{95} + 7 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 1.00000 + 1.73205i 2.00000 0 1.50000 + 2.59808i 0 −0.500000 0.866025i 0
133.1 0 0.500000 + 0.866025i 1.00000 1.73205i 2.00000 0 1.50000 2.59808i 0 −0.500000 + 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.i.b 2
13.c even 3 1 inner 429.2.i.b 2
13.c even 3 1 5577.2.a.d 1
13.e even 6 1 5577.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.b 2 1.a even 1 1 trivial
429.2.i.b 2 13.c even 3 1 inner
5577.2.a.c 1 13.e even 6 1
5577.2.a.d 1 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 14)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$73$ \( (T + 9)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$97$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
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