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Properties

Label 672.4.b.a
Level $672$
Weight $4$
Character orbit 672.b
Analytic conductor $39.649$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,4,Mod(223,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.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: no
Analytic conductor: \(39.6492835239\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 72 q^{3} - 20 q^{7} + 216 q^{9} + 56 q^{19} + 60 q^{21} - 432 q^{25} - 648 q^{27} + 464 q^{31} - 568 q^{35} + 504 q^{37} + 560 q^{47} - 128 q^{49} - 784 q^{53} + 424 q^{55} - 168 q^{57} - 800 q^{59} - 180 q^{63} + 560 q^{65} + 1296 q^{75} - 1568 q^{77} + 1944 q^{81} - 1936 q^{83} + 3000 q^{85} + 496 q^{91} - 1392 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 0 −3.00000 0 − 19.8256i 0 13.1775 − 13.0136i 0 9.00000 0
223.2 0 −3.00000 0 − 15.7575i 0 15.0622 + 10.7764i 0 9.00000 0
223.3 0 −3.00000 0 − 15.0520i 0 −17.2151 − 6.82941i 0 9.00000 0
223.4 0 −3.00000 0 − 14.9540i 0 −18.3556 + 2.46445i 0 9.00000 0
223.5 0 −3.00000 0 − 13.1053i 0 −9.87442 − 15.6683i 0 9.00000 0
223.6 0 −3.00000 0 − 12.7173i 0 −2.54263 + 18.3449i 0 9.00000 0
223.7 0 −3.00000 0 − 12.4907i 0 8.98608 − 16.1941i 0 9.00000 0
223.8 0 −3.00000 0 − 8.63807i 0 −18.2735 − 3.01325i 0 9.00000 0
223.9 0 −3.00000 0 − 5.77465i 0 −3.77600 + 18.1312i 0 9.00000 0
223.10 0 −3.00000 0 − 4.15525i 0 10.5863 − 15.1964i 0 9.00000 0
223.11 0 −3.00000 0 − 2.26482i 0 −5.56016 + 17.6659i 0 9.00000 0
223.12 0 −3.00000 0 − 2.14586i 0 17.7853 − 5.16554i 0 9.00000 0
223.13 0 −3.00000 0 2.14586i 0 17.7853 + 5.16554i 0 9.00000 0
223.14 0 −3.00000 0 2.26482i 0 −5.56016 − 17.6659i 0 9.00000 0
223.15 0 −3.00000 0 4.15525i 0 10.5863 + 15.1964i 0 9.00000 0
223.16 0 −3.00000 0 5.77465i 0 −3.77600 − 18.1312i 0 9.00000 0
223.17 0 −3.00000 0 8.63807i 0 −18.2735 + 3.01325i 0 9.00000 0
223.18 0 −3.00000 0 12.4907i 0 8.98608 + 16.1941i 0 9.00000 0
223.19 0 −3.00000 0 12.7173i 0 −2.54263 − 18.3449i 0 9.00000 0
223.20 0 −3.00000 0 13.1053i 0 −9.87442 + 15.6683i 0 9.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.b.a ✓ 24
4.b odd 2 1 672.4.b.b yes 24
7.b odd 2 1 672.4.b.b yes 24
8.b even 2 1 1344.4.b.j 24
8.d odd 2 1 1344.4.b.i 24
28.d even 2 1 inner 672.4.b.a ✓ 24
56.e even 2 1 1344.4.b.j 24
56.h odd 2 1 1344.4.b.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.b.a ✓ 24 1.a even 1 1 trivial
672.4.b.a ✓ 24 28.d even 2 1 inner
672.4.b.b yes 24 4.b odd 2 1
672.4.b.b yes 24 7.b odd 2 1
1344.4.b.i 24 8.d odd 2 1
1344.4.b.i 24 56.h odd 2 1
1344.4.b.j 24 8.b even 2 1
1344.4.b.j 24 56.e even 2 1