Properties

Label 4.13.b.a
Level $4$
Weight $13$
Character orbit 4.b
Self dual yes
Analytic conductor $3.656$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(3.65597526911\)
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)
 
\(f(q)\) \(=\) \( q - 64 q^{2} + 4096 q^{4} + 23506 q^{5} - 262144 q^{8} + 531441 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 64 q^{2} + 4096 q^{4} + 23506 q^{5} - 262144 q^{8} + 531441 q^{9} - 1504384 q^{10} + 6911282 q^{13} + 16777216 q^{16} - 47295038 q^{17} - 34012224 q^{18} + 96280576 q^{20} + 308391411 q^{25} - 442322048 q^{26} - 173439758 q^{29} - 1073741824 q^{32} + 3026882432 q^{34} + 2176782336 q^{36} - 2050092718 q^{37} - 6161956864 q^{40} - 2285065118 q^{41} + 12492052146 q^{45} + 13841287201 q^{49} - 19737050304 q^{50} + 28308611072 q^{52} - 43462597358 q^{53} + 11100144512 q^{58} - 47844884878 q^{61} + 68719476736 q^{64} + 162456594692 q^{65} - 193720475648 q^{68} - 139314069504 q^{72} - 119852347678 q^{73} + 131205933952 q^{74} + 394365239296 q^{80} + 282429536481 q^{81} + 146244167552 q^{82} - 1111717163228 q^{85} + 907573615522 q^{89} - 799491337344 q^{90} + 502341690242 q^{97} - 885842380864 q^{98}+O(q^{100}) \) 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
−64.0000 0 4096.00 23506.0 0 0 −262144. 531441. −1.50438e6
\(n\): 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 4.13.b.a 1
3.b odd 2 1 36.13.d.a 1
4.b odd 2 1 CM 4.13.b.a 1
8.b even 2 1 64.13.c.a 1
8.d odd 2 1 64.13.c.a 1
12.b even 2 1 36.13.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.13.b.a 1 1.a even 1 1 trivial
4.13.b.a 1 4.b odd 2 1 CM
36.13.d.a 1 3.b odd 2 1
36.13.d.a 1 12.b even 2 1
64.13.c.a 1 8.b even 2 1
64.13.c.a 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{13}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 64 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 23506 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 6911282 \) Copy content Toggle raw display
$17$ \( T + 47295038 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 173439758 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 2050092718 \) Copy content Toggle raw display
$41$ \( T + 2285065118 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 43462597358 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 47844884878 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 119852347678 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 907573615522 \) Copy content Toggle raw display
$97$ \( T - 502341690242 \) Copy content Toggle raw display
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