Properties

Label 435.3.h.b
Level $435$
Weight $3$
Character orbit 435.h
Analytic conductor $11.853$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,3,Mod(86,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.86");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 435.h (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: \(11.8528914997\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta + 2) q^{3} + \beta q^{5} + ( - 2 \beta + 4) q^{6} + 8 q^{7} - 8 q^{8} + ( - 4 \beta - 1) q^{9} + 2 \beta q^{10} + 11 q^{11} + 2 q^{13} + 16 q^{14} + (2 \beta + 5) q^{15} - 16 q^{16}+ \cdots + ( - 44 \beta - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 4 q^{3} + 8 q^{6} + 16 q^{7} - 16 q^{8} - 2 q^{9} + 22 q^{11} + 4 q^{13} + 32 q^{14} + 10 q^{15} - 32 q^{16} + 64 q^{17} - 4 q^{18} + 32 q^{21} + 44 q^{22} - 32 q^{24} - 10 q^{25} + 8 q^{26}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/435\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(146\) \(262\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1
2.23607i
2.23607i
2.00000 2.00000 2.23607i 0 2.23607i 4.00000 4.47214i 8.00000 −8.00000 −1.00000 8.94427i 4.47214i
86.2 2.00000 2.00000 + 2.23607i 0 2.23607i 4.00000 + 4.47214i 8.00000 −8.00000 −1.00000 + 8.94427i 4.47214i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
87.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.3.h.b yes 2
3.b odd 2 1 435.3.h.a 2
29.b even 2 1 435.3.h.a 2
87.d odd 2 1 inner 435.3.h.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.3.h.a 2 3.b odd 2 1
435.3.h.a 2 29.b even 2 1
435.3.h.b yes 2 1.a even 1 1 trivial
435.3.h.b yes 2 87.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{3}^{\mathrm{new}}(435, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 4T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( (T - 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( T^{2} + 1125 \) Copy content Toggle raw display
$29$ \( (T + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 720 \) Copy content Toggle raw display
$37$ \( T^{2} + 1125 \) Copy content Toggle raw display
$41$ \( (T + 37)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2205 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 405 \) Copy content Toggle raw display
$59$ \( T^{2} + 4500 \) Copy content Toggle raw display
$61$ \( T^{2} + 180 \) Copy content Toggle raw display
$67$ \( (T - 38)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8820 \) Copy content Toggle raw display
$73$ \( T^{2} + 5445 \) Copy content Toggle raw display
$79$ \( T^{2} + 180 \) Copy content Toggle raw display
$83$ \( T^{2} + 2205 \) Copy content Toggle raw display
$89$ \( (T + 166)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 19845 \) Copy content Toggle raw display
show more
show less