Properties

Label 624.4.n.c
Level $624$
Weight $4$
Character orbit 624.n
Analytic conductor $36.817$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(623,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.623");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.n (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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 172 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 172 q^{9} + 96 q^{13} + 1392 q^{25} - 1152 q^{49} - 96 q^{61} + 2640 q^{69} - 5396 q^{81}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
623.1 0 −4.94584 1.59332i 0 −2.47960 0 21.7201 0 21.9227 + 15.7606i 0
623.2 0 −4.94584 1.59332i 0 2.47960 0 −21.7201 0 21.9227 + 15.7606i 0
623.3 0 −4.94584 + 1.59332i 0 −2.47960 0 21.7201 0 21.9227 15.7606i 0
623.4 0 −4.94584 + 1.59332i 0 2.47960 0 −21.7201 0 21.9227 15.7606i 0
623.5 0 −4.38900 2.78149i 0 −18.3912 0 7.53270 0 11.5266 + 24.4159i 0
623.6 0 −4.38900 2.78149i 0 18.3912 0 −7.53270 0 11.5266 + 24.4159i 0
623.7 0 −4.38900 + 2.78149i 0 −18.3912 0 7.53270 0 11.5266 24.4159i 0
623.8 0 −4.38900 + 2.78149i 0 18.3912 0 −7.53270 0 11.5266 24.4159i 0
623.9 0 −2.74324 4.41301i 0 −14.3044 0 −18.8811 0 −11.9493 + 24.2119i 0
623.10 0 −2.74324 4.41301i 0 14.3044 0 18.8811 0 −11.9493 + 24.2119i 0
623.11 0 −2.74324 + 4.41301i 0 −14.3044 0 −18.8811 0 −11.9493 24.2119i 0
623.12 0 −2.74324 + 4.41301i 0 14.3044 0 18.8811 0 −11.9493 24.2119i 0
623.13 0 2.74324 4.41301i 0 −14.3044 0 18.8811 0 −11.9493 24.2119i 0
623.14 0 2.74324 4.41301i 0 14.3044 0 −18.8811 0 −11.9493 24.2119i 0
623.15 0 2.74324 + 4.41301i 0 −14.3044 0 18.8811 0 −11.9493 + 24.2119i 0
623.16 0 2.74324 + 4.41301i 0 14.3044 0 −18.8811 0 −11.9493 + 24.2119i 0
623.17 0 4.38900 2.78149i 0 −18.3912 0 −7.53270 0 11.5266 24.4159i 0
623.18 0 4.38900 2.78149i 0 18.3912 0 7.53270 0 11.5266 24.4159i 0
623.19 0 4.38900 + 2.78149i 0 −18.3912 0 −7.53270 0 11.5266 + 24.4159i 0
623.20 0 4.38900 + 2.78149i 0 18.3912 0 7.53270 0 11.5266 + 24.4159i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 623.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner
156.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.n.c 24
3.b odd 2 1 inner 624.4.n.c 24
4.b odd 2 1 inner 624.4.n.c 24
12.b even 2 1 inner 624.4.n.c 24
13.b even 2 1 inner 624.4.n.c 24
39.d odd 2 1 inner 624.4.n.c 24
52.b odd 2 1 inner 624.4.n.c 24
156.h even 2 1 inner 624.4.n.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.4.n.c 24 1.a even 1 1 trivial
624.4.n.c 24 3.b odd 2 1 inner
624.4.n.c 24 4.b odd 2 1 inner
624.4.n.c 24 12.b even 2 1 inner
624.4.n.c 24 13.b even 2 1 inner
624.4.n.c 24 39.d odd 2 1 inner
624.4.n.c 24 52.b odd 2 1 inner
624.4.n.c 24 156.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{6} - 549T_{5}^{4} + 72546T_{5}^{2} - 425520 \) Copy content Toggle raw display
\( T_{7}^{6} - 885T_{7}^{4} + 215178T_{7}^{2} - 9542880 \) Copy content Toggle raw display