Properties

Label 429.2.e.a
Level $429$
Weight $2$
Character orbit 429.e
Analytic conductor $3.426$
Analytic rank $0$
Dimension $8$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [429,2,Mod(428,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.428");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.e (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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.151613669376.21
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{2} q^{3} + ( - \beta_{2} - 2) q^{4} + ( - \beta_{6} + \beta_1) q^{5} + (\beta_{6} - \beta_{3} + \beta_1) q^{6} + (\beta_{6} - \beta_{3} + \beta_1) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{2} q^{3} + ( - \beta_{2} - 2) q^{4} + ( - \beta_{6} + \beta_1) q^{5} + (\beta_{6} - \beta_{3} + \beta_1) q^{6} + (\beta_{6} - \beta_{3} + \beta_1) q^{8} + 3 q^{9} + ( - 2 \beta_{7} + 2 \beta_{5} - \beta_{2} + 1) q^{10} + (\beta_{4} + \beta_{3} - \beta_1) q^{11} + (2 \beta_{2} + 3) q^{12} + \beta_{5} q^{13} + (\beta_{6} - 2 \beta_{4} - \beta_1) q^{15} + (2 \beta_{2} - 1) q^{16} + 3 \beta_{3} q^{18} + (3 \beta_{6} - 2 \beta_{4} - 3 \beta_1) q^{20} + (\beta_{7} - 2 \beta_{2} - 4) q^{22} + 3 \beta_{3} q^{24} + (4 \beta_{2} + 5) q^{25} + (\beta_{6} - 3 \beta_{4} - \beta_1) q^{26} - 3 \beta_{2} q^{27} + (2 \beta_{7} - 4 \beta_{5} + \beta_{2} - 1) q^{30} - \beta_{3} q^{32} + (\beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{33} + ( - 3 \beta_{2} - 6) q^{36} + (2 \beta_{7} - \beta_{5} + \beta_{2} - 1) q^{39} + (2 \beta_{7} - 4 \beta_{5} + \beta_{2} - 1) q^{40} + ( - \beta_{6} - 4 \beta_{3} - \beta_1) q^{41} + ( - 4 \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + 2) q^{43} + (\beta_{6} - 4 \beta_{3} + 2 \beta_1) q^{44} + ( - 3 \beta_{6} + 3 \beta_1) q^{45} + (3 \beta_{6} + 2 \beta_{4} - 3 \beta_1) q^{47} + (\beta_{2} - 6) q^{48} - 7 q^{49} + ( - 4 \beta_{6} + 9 \beta_{3} - 4 \beta_1) q^{50} + (2 \beta_{7} - 3 \beta_{5} + \beta_{2} - 1) q^{52} + (3 \beta_{6} - 3 \beta_{3} + 3 \beta_1) q^{54} + ( - 2 \beta_{7} + 3 \beta_{5} - 3) q^{55} + ( - 3 \beta_{6} + 2 \beta_{4} + 3 \beta_1) q^{59} + ( - 5 \beta_{6} + 4 \beta_{4} + 5 \beta_1) q^{60} + 2 \beta_{5} q^{61} + (5 \beta_{2} + 2) q^{64} + ( - 3 \beta_{6} + 4 \beta_{3} - 3 \beta_1) q^{65} + (\beta_{7} + \beta_{5} + 4 \beta_{2} + 7) q^{66} + ( - \beta_{6} - 6 \beta_{4} + \beta_1) q^{71} + (3 \beta_{6} - 3 \beta_{3} + 3 \beta_1) q^{72} + ( - 5 \beta_{2} - 12) q^{75} + ( - 4 \beta_{6} - \beta_{4} + 4 \beta_1) q^{78} - 4 \beta_{5} q^{79} + ( - \beta_{6} + 4 \beta_{4} + \beta_1) q^{80} + 9 q^{81} + (\beta_{2} + 17) q^{82} + ( - \beta_{6} - 6 \beta_{3} - \beta_1) q^{83} + (8 \beta_{6} + 2 \beta_{4} - 8 \beta_1) q^{86} + (\beta_{7} + \beta_{5} + 4 \beta_{2} + 7) q^{88} + ( - 3 \beta_{6} + 4 \beta_{4} + 3 \beta_1) q^{89} + ( - 6 \beta_{7} + 6 \beta_{5} - 3 \beta_{2} + 3) q^{90} + (6 \beta_{7} - 4 \beta_{5} + 3 \beta_{2} - 3) q^{94} + ( - \beta_{6} + \beta_{3} - \beta_1) q^{96} - 7 \beta_{3} q^{98} + (3 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 24 q^{9} + 24 q^{12} - 8 q^{16} - 28 q^{22} + 40 q^{25} - 48 q^{36} - 48 q^{48} - 56 q^{49} - 32 q^{55} + 16 q^{64} + 60 q^{66} - 96 q^{75} + 72 q^{81} + 136 q^{82} + 60 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 5\nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 5\nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 9\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 3\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{4} + \nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} - \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - \beta_{5} + \beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{4} - 3\beta_{3} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{5} + 9\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -10\beta_{6} - 3\beta_{4} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
428.1
−0.752986 1.19709i
0.752986 1.19709i
1.19709 0.752986i
−1.19709 0.752986i
1.19709 + 0.752986i
−1.19709 + 0.752986i
−0.752986 + 1.19709i
0.752986 + 1.19709i
2.39417i −1.73205 −3.73205 −4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.2 2.39417i −1.73205 −3.73205 4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.3 1.50597i 1.73205 −0.267949 −1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.4 1.50597i 1.73205 −0.267949 1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.5 1.50597i 1.73205 −0.267949 −1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.6 1.50597i 1.73205 −0.267949 1.75265 2.60842i 0 2.60842i 3.00000 2.63945i
428.7 2.39417i −1.73205 −3.73205 −4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
428.8 2.39417i −1.73205 −3.73205 4.11439 4.14682i 0 4.14682i 3.00000 9.85055i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 428.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 2 1 inner
143.d odd 2 1 inner
429.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.e.a 8
3.b odd 2 1 inner 429.2.e.a 8
11.b odd 2 1 inner 429.2.e.a 8
13.b even 2 1 inner 429.2.e.a 8
33.d even 2 1 inner 429.2.e.a 8
39.d odd 2 1 CM 429.2.e.a 8
143.d odd 2 1 inner 429.2.e.a 8
429.e even 2 1 inner 429.2.e.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.e.a 8 1.a even 1 1 trivial
429.2.e.a 8 3.b odd 2 1 inner
429.2.e.a 8 11.b odd 2 1 inner
429.2.e.a 8 13.b even 2 1 inner
429.2.e.a 8 33.d even 2 1 inner
429.2.e.a 8 39.d odd 2 1 CM
429.2.e.a 8 143.d odd 2 1 inner
429.2.e.a 8 429.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 8T_{2}^{2} + 13 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 8 T^{2} + 13)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 20 T^{2} + 52)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 190 T^{4} + 14641 \) Copy content Toggle raw display
$13$ \( (T^{2} + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 164 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 156)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 188 T^{2} + 8788)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 236 T^{2} + 52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 52)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 284 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 208)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 332 T^{2} + 27508)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 356 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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