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Properties

Label 624.4.d.c
Level $624$
Weight $4$
Character orbit 624.d
Analytic conductor $36.817$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(287,624)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.287");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.d (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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 84 q^{9} - 312 q^{13} - 300 q^{21} - 240 q^{25} + 240 q^{33} - 456 q^{37} + 1836 q^{45} - 1632 q^{49} + 168 q^{57} - 960 q^{61} + 2760 q^{69} + 1248 q^{73} - 468 q^{81} - 7704 q^{85} + 3336 q^{93}+ \cdots - 2496 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1 0 −5.00421 − 1.39923i 0 − 1.15858i 0 9.33514i 0 23.0843 + 14.0041i 0
287.2 0 −5.00421 + 1.39923i 0 1.15858i 0 − 9.33514i 0 23.0843 − 14.0041i 0
287.3 0 −4.62561 − 2.36721i 0 − 8.18552i 0 30.2381i 0 15.7926 + 21.8996i 0
287.4 0 −4.62561 + 2.36721i 0 8.18552i 0 − 30.2381i 0 15.7926 − 21.8996i 0
287.5 0 −3.81974 − 3.52273i 0 − 18.3557i 0 − 21.7737i 0 2.18081 + 26.9118i 0
287.6 0 −3.81974 + 3.52273i 0 18.3557i 0 21.7737i 0 2.18081 − 26.9118i 0
287.7 0 −2.35002 − 4.63437i 0 − 6.29951i 0 − 30.3725i 0 −15.9549 + 21.7817i 0
287.8 0 −2.35002 + 4.63437i 0 6.29951i 0 30.3725i 0 −15.9549 − 21.7817i 0
287.9 0 −1.92666 − 4.82576i 0 18.6497i 0 4.79314i 0 −19.5759 + 18.5952i 0
287.10 0 −1.92666 + 4.82576i 0 − 18.6497i 0 − 4.79314i 0 −19.5759 − 18.5952i 0
287.11 0 −0.486351 − 5.17334i 0 4.15078i 0 6.70503i 0 −26.5269 + 5.03212i 0
287.12 0 −0.486351 + 5.17334i 0 − 4.15078i 0 − 6.70503i 0 −26.5269 − 5.03212i 0
287.13 0 0.486351 − 5.17334i 0 − 4.15078i 0 6.70503i 0 −26.5269 − 5.03212i 0
287.14 0 0.486351 + 5.17334i 0 4.15078i 0 − 6.70503i 0 −26.5269 + 5.03212i 0
287.15 0 1.92666 − 4.82576i 0 − 18.6497i 0 4.79314i 0 −19.5759 − 18.5952i 0
287.16 0 1.92666 + 4.82576i 0 18.6497i 0 − 4.79314i 0 −19.5759 + 18.5952i 0
287.17 0 2.35002 − 4.63437i 0 6.29951i 0 − 30.3725i 0 −15.9549 − 21.7817i 0
287.18 0 2.35002 + 4.63437i 0 − 6.29951i 0 30.3725i 0 −15.9549 + 21.7817i 0
287.19 0 3.81974 − 3.52273i 0 18.3557i 0 − 21.7737i 0 2.18081 − 26.9118i 0
287.20 0 3.81974 + 3.52273i 0 − 18.3557i 0 21.7737i 0 2.18081 + 26.9118i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.24
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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 810T_{5}^{10} + 207621T_{5}^{8} + 17923788T_{5}^{6} + 582053364T_{5}^{4} + 6117946848T_{5}^{2} + 7206159168 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display