Properties

Label 435.3.b.b
Level $435$
Weight $3$
Character orbit 435.b
Self dual yes
Analytic conductor $11.853$
Analytic rank $0$
Dimension $1$
CM discriminant -435
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,3,Mod(434,435)]
 
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, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.434");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 435.b (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: yes
Analytic conductor: \(11.8528914997\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} + 4 q^{4} + 5 q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 4 q^{4} + 5 q^{5} + 9 q^{9} + 7 q^{11} - 12 q^{12} - 15 q^{15} + 16 q^{16} + 20 q^{20} - 41 q^{23} + 25 q^{25} - 27 q^{27} + 29 q^{29} - 21 q^{33} + 36 q^{36} + 71 q^{37} - 53 q^{41} + 59 q^{43} + 28 q^{44} + 45 q^{45} - 48 q^{48} + 49 q^{49} + 19 q^{53} + 35 q^{55} - 60 q^{60} + 64 q^{64} + 123 q^{69} - q^{73} - 75 q^{75} + 80 q^{80} + 81 q^{81} + 79 q^{83} - 87 q^{87} - 62 q^{89} - 164 q^{92} - 49 q^{97} + 63 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
434.1
0
0 −3.00000 4.00000 5.00000 0 0 0 9.00000 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
435.b odd 2 1 CM by \(\Q(\sqrt{-435}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.3.b.b yes 1
3.b odd 2 1 435.3.b.a 1
5.b even 2 1 435.3.b.c yes 1
15.d odd 2 1 435.3.b.d yes 1
29.b even 2 1 435.3.b.d yes 1
87.d odd 2 1 435.3.b.c yes 1
145.d even 2 1 435.3.b.a 1
435.b odd 2 1 CM 435.3.b.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.3.b.a 1 3.b odd 2 1
435.3.b.a 1 145.d even 2 1
435.3.b.b yes 1 1.a even 1 1 trivial
435.3.b.b yes 1 435.b odd 2 1 CM
435.3.b.c yes 1 5.b even 2 1
435.3.b.c yes 1 87.d odd 2 1
435.3.b.d yes 1 15.d odd 2 1
435.3.b.d yes 1 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(435, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{11} - 7 \) Copy content Toggle raw display
\( T_{23} + 41 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 7 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 41 \) Copy content Toggle raw display
$29$ \( T - 29 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 71 \) Copy content Toggle raw display
$41$ \( T + 53 \) Copy content Toggle raw display
$43$ \( T - 59 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 19 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 1 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 79 \) Copy content Toggle raw display
$89$ \( T + 62 \) Copy content Toggle raw display
$97$ \( T + 49 \) Copy content Toggle raw display
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