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Properties

Label 624.4.d.d
Level $624$
Weight $4$
Character orbit 624.d
Analytic conductor $36.817$
Analytic rank $0$
Dimension $24$
CM no
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{9} + 312 q^{13} + 284 q^{21} - 48 q^{25} - 848 q^{33} + 648 q^{37} + 1108 q^{45} - 288 q^{49} - 888 q^{57} - 192 q^{61} - 1192 q^{69} + 2496 q^{73} - 788 q^{81} + 1272 q^{85} - 6136 q^{93} + 7392 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −5.19386 − 0.154282i 0 12.3762i 0 15.8753i 0 26.9524 + 1.60264i 0
287.2 0 −5.19386 + 0.154282i 0 − 12.3762i 0 − 15.8753i 0 26.9524 − 1.60264i 0
287.3 0 −4.90634 − 1.71110i 0 8.87899i 0 − 19.8038i 0 21.1443 + 16.7905i 0
287.4 0 −4.90634 + 1.71110i 0 − 8.87899i 0 19.8038i 0 21.1443 − 16.7905i 0
287.5 0 −3.44279 − 3.89194i 0 − 13.2715i 0 − 5.95848i 0 −3.29443 + 26.7983i 0
287.6 0 −3.44279 + 3.89194i 0 13.2715i 0 5.95848i 0 −3.29443 − 26.7983i 0
287.7 0 −2.90133 − 4.31072i 0 9.14159i 0 − 18.5185i 0 −10.1646 + 25.0136i 0
287.8 0 −2.90133 + 4.31072i 0 − 9.14159i 0 18.5185i 0 −10.1646 − 25.0136i 0
287.9 0 −2.22958 − 4.69350i 0 − 16.3635i 0 31.0339i 0 −17.0579 + 20.9291i 0
287.10 0 −2.22958 + 4.69350i 0 16.3635i 0 − 31.0339i 0 −17.0579 − 20.9291i 0
287.11 0 −1.48665 − 4.97894i 0 1.59033i 0 12.0101i 0 −22.5797 + 14.8039i 0
287.12 0 −1.48665 + 4.97894i 0 − 1.59033i 0 − 12.0101i 0 −22.5797 − 14.8039i 0
287.13 0 1.48665 − 4.97894i 0 − 1.59033i 0 12.0101i 0 −22.5797 − 14.8039i 0
287.14 0 1.48665 + 4.97894i 0 1.59033i 0 − 12.0101i 0 −22.5797 + 14.8039i 0
287.15 0 2.22958 − 4.69350i 0 16.3635i 0 31.0339i 0 −17.0579 − 20.9291i 0
287.16 0 2.22958 + 4.69350i 0 − 16.3635i 0 − 31.0339i 0 −17.0579 + 20.9291i 0
287.17 0 2.90133 − 4.31072i 0 − 9.14159i 0 − 18.5185i 0 −10.1646 − 25.0136i 0
287.18 0 2.90133 + 4.31072i 0 9.14159i 0 18.5185i 0 −10.1646 + 25.0136i 0
287.19 0 3.44279 − 3.89194i 0 13.2715i 0 − 5.95848i 0 −3.29443 − 26.7983i 0
287.20 0 3.44279 + 3.89194i 0 − 13.2715i 0 5.95848i 0 −3.29443 + 26.7983i 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.d ✓ 24
3.b odd 2 1 inner 624.4.d.d ✓ 24
4.b odd 2 1 inner 624.4.d.d ✓ 24
12.b even 2 1 inner 624.4.d.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.4.d.d ✓ 24 1.a even 1 1 trivial
624.4.d.d ✓ 24 3.b odd 2 1 inner
624.4.d.d ✓ 24 4.b odd 2 1 inner
624.4.d.d ✓ 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 762T_{5}^{10} + 220629T_{5}^{8} + 30412060T_{5}^{6} + 2007358500T_{5}^{4} + 52478110464T_{5}^{2} + 120367756288 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display