Properties

Label 4.5.b.a
Level $4$
Weight $5$
Character orbit 4.b
Self dual yes
Analytic conductor $0.413$
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] = [4,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(0.413479852335\)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} - 14 q^{5} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} - 14 q^{5} - 64 q^{8} + 81 q^{9} + 56 q^{10} - 238 q^{13} + 256 q^{16} + 322 q^{17} - 324 q^{18} - 224 q^{20} - 429 q^{25} + 952 q^{26} + 82 q^{29} - 1024 q^{32} - 1288 q^{34} + 1296 q^{36} + 2162 q^{37} + 896 q^{40} - 3038 q^{41} - 1134 q^{45} + 2401 q^{49} + 1716 q^{50} - 3808 q^{52} + 2482 q^{53} - 328 q^{58} - 6958 q^{61} + 4096 q^{64} + 3332 q^{65} + 5152 q^{68} - 5184 q^{72} + 1442 q^{73} - 8648 q^{74} - 3584 q^{80} + 6561 q^{81} + 12152 q^{82} - 4508 q^{85} - 9758 q^{89} + 4536 q^{90} - 1918 q^{97} - 9604 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \eta(z)^{4}\eta(2z)^{2}\eta(4z)^{4}=q\prod_{n=1}^\infty(1 - q^{n})^{4}(1 - q^{2n})^{2}(1 - q^{4n})^{4}\)

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
−4.00000 0 16.0000 −14.0000 0 0 −64.0000 81.0000 56.0000
\(n\): 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 4.5.b.a 1
3.b odd 2 1 36.5.d.a 1
4.b odd 2 1 CM 4.5.b.a 1
5.b even 2 1 100.5.b.a 1
5.c odd 4 2 100.5.d.a 2
7.b odd 2 1 196.5.c.a 1
8.b even 2 1 64.5.c.a 1
8.d odd 2 1 64.5.c.a 1
12.b even 2 1 36.5.d.a 1
16.e even 4 2 256.5.d.c 2
16.f odd 4 2 256.5.d.c 2
20.d odd 2 1 100.5.b.a 1
20.e even 4 2 100.5.d.a 2
24.f even 2 1 576.5.g.b 1
24.h odd 2 1 576.5.g.b 1
28.d even 2 1 196.5.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.5.b.a 1 1.a even 1 1 trivial
4.5.b.a 1 4.b odd 2 1 CM
36.5.d.a 1 3.b odd 2 1
36.5.d.a 1 12.b even 2 1
64.5.c.a 1 8.b even 2 1
64.5.c.a 1 8.d odd 2 1
100.5.b.a 1 5.b even 2 1
100.5.b.a 1 20.d odd 2 1
100.5.d.a 2 5.c odd 4 2
100.5.d.a 2 20.e even 4 2
196.5.c.a 1 7.b odd 2 1
196.5.c.a 1 28.d even 2 1
256.5.d.c 2 16.e even 4 2
256.5.d.c 2 16.f odd 4 2
576.5.g.b 1 24.f even 2 1
576.5.g.b 1 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 14 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 238 \) Copy content Toggle raw display
$17$ \( T - 322 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 82 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 2162 \) Copy content Toggle raw display
$41$ \( T + 3038 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 2482 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 6958 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 1442 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 9758 \) Copy content Toggle raw display
$97$ \( T + 1918 \) Copy content Toggle raw display
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