Properties

Label 4.9.b.a
Level $4$
Weight $9$
Character orbit 4.b
Self dual yes
Analytic conductor $1.630$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,9,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(1.62951444024\)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} + 256 q^{4} - 1054 q^{5} + 4096 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} + 256 q^{4} - 1054 q^{5} + 4096 q^{8} + 6561 q^{9} - 16864 q^{10} - 478 q^{13} + 65536 q^{16} - 63358 q^{17} + 104976 q^{18} - 269824 q^{20} + 720291 q^{25} - 7648 q^{26} - 1407838 q^{29} + 1048576 q^{32} - 1013728 q^{34} + 1679616 q^{36} + 925922 q^{37} - 4317184 q^{40} + 3577922 q^{41} - 6915294 q^{45} + 5764801 q^{49} + 11524656 q^{50} - 122368 q^{52} - 9620638 q^{53} - 22525408 q^{58} + 20722082 q^{61} + 16777216 q^{64} + 503812 q^{65} - 16219648 q^{68} + 26873856 q^{72} - 54717118 q^{73} + 14814752 q^{74} - 69074944 q^{80} + 43046721 q^{81} + 57246752 q^{82} + 66779332 q^{85} - 30265918 q^{89} - 110644704 q^{90} - 173379838 q^{97} + 92236816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
16.0000 0 256.000 −1054.00 0 0 4096.00 6561.00 −16864.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.9.b.a 1
3.b odd 2 1 36.9.d.a 1
4.b odd 2 1 CM 4.9.b.a 1
5.b even 2 1 100.9.b.a 1
5.c odd 4 2 100.9.d.a 2
8.b even 2 1 64.9.c.a 1
8.d odd 2 1 64.9.c.a 1
12.b even 2 1 36.9.d.a 1
16.e even 4 2 256.9.d.a 2
16.f odd 4 2 256.9.d.a 2
20.d odd 2 1 100.9.b.a 1
20.e even 4 2 100.9.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.a 1 1.a even 1 1 trivial
4.9.b.a 1 4.b odd 2 1 CM
36.9.d.a 1 3.b odd 2 1
36.9.d.a 1 12.b even 2 1
64.9.c.a 1 8.b even 2 1
64.9.c.a 1 8.d odd 2 1
100.9.b.a 1 5.b even 2 1
100.9.b.a 1 20.d odd 2 1
100.9.d.a 2 5.c odd 4 2
100.9.d.a 2 20.e even 4 2
256.9.d.a 2 16.e even 4 2
256.9.d.a 2 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{9}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1054 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 478 \) Copy content Toggle raw display
$17$ \( T + 63358 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 1407838 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 925922 \) Copy content Toggle raw display
$41$ \( T - 3577922 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 9620638 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 20722082 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 54717118 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 30265918 \) Copy content Toggle raw display
$97$ \( T + 173379838 \) Copy content Toggle raw display
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