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Properties

Label 672.4.k.e
Level $672$
Weight $4$
Character orbit 672.k
Analytic conductor $39.649$
Analytic rank $0$
Dimension $48$
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] = [672,4,Mod(545,672)]
 
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([0, 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("672.545");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.k (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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 56 q^{9} + 64 q^{21} + 960 q^{25} - 1056 q^{37} - 1056 q^{49} - 200 q^{57} - 3456 q^{81} + 1344 q^{85} + 3120 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1 0 −5.13847 − 0.772120i 0 −3.38040 0 −16.8601 + 7.66407i 0 25.8077 + 7.93502i 0
545.2 0 −5.13847 − 0.772120i 0 3.38040 0 16.8601 + 7.66407i 0 25.8077 + 7.93502i 0
545.3 0 −5.13847 + 0.772120i 0 −3.38040 0 −16.8601 − 7.66407i 0 25.8077 − 7.93502i 0
545.4 0 −5.13847 + 0.772120i 0 3.38040 0 16.8601 − 7.66407i 0 25.8077 − 7.93502i 0
545.5 0 −4.62359 − 2.37116i 0 −19.9503 0 14.5334 − 11.4795i 0 15.7552 + 21.9265i 0
545.6 0 −4.62359 − 2.37116i 0 19.9503 0 −14.5334 − 11.4795i 0 15.7552 + 21.9265i 0
545.7 0 −4.62359 + 2.37116i 0 −19.9503 0 14.5334 + 11.4795i 0 15.7552 − 21.9265i 0
545.8 0 −4.62359 + 2.37116i 0 19.9503 0 −14.5334 + 11.4795i 0 15.7552 − 21.9265i 0
545.9 0 −4.20092 − 3.05815i 0 −11.8554 0 −3.80545 − 18.1251i 0 8.29548 + 25.6941i 0
545.10 0 −4.20092 − 3.05815i 0 11.8554 0 3.80545 − 18.1251i 0 8.29548 + 25.6941i 0
545.11 0 −4.20092 + 3.05815i 0 −11.8554 0 −3.80545 + 18.1251i 0 8.29548 − 25.6941i 0
545.12 0 −4.20092 + 3.05815i 0 11.8554 0 3.80545 + 18.1251i 0 8.29548 − 25.6941i 0
545.13 0 −3.72018 − 3.62770i 0 −0.917468 0 −11.0470 + 14.8648i 0 0.679529 + 26.9914i 0
545.14 0 −3.72018 − 3.62770i 0 0.917468 0 11.0470 + 14.8648i 0 0.679529 + 26.9914i 0
545.15 0 −3.72018 + 3.62770i 0 −0.917468 0 −11.0470 − 14.8648i 0 0.679529 − 26.9914i 0
545.16 0 −3.72018 + 3.62770i 0 0.917468 0 11.0470 − 14.8648i 0 0.679529 − 26.9914i 0
545.17 0 −2.20064 − 4.70714i 0 −15.2263 0 −8.40218 + 16.5046i 0 −17.3144 + 20.7175i 0
545.18 0 −2.20064 − 4.70714i 0 15.2263 0 8.40218 + 16.5046i 0 −17.3144 + 20.7175i 0
545.19 0 −2.20064 + 4.70714i 0 −15.2263 0 −8.40218 − 16.5046i 0 −17.3144 − 20.7175i 0
545.20 0 −2.20064 + 4.70714i 0 15.2263 0 8.40218 − 16.5046i 0 −17.3144 − 20.7175i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.48
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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.k.e ✓ 48
3.b odd 2 1 inner 672.4.k.e ✓ 48
4.b odd 2 1 inner 672.4.k.e ✓ 48
7.b odd 2 1 inner 672.4.k.e ✓ 48
12.b even 2 1 inner 672.4.k.e ✓ 48
21.c even 2 1 inner 672.4.k.e ✓ 48
28.d even 2 1 inner 672.4.k.e ✓ 48
84.h odd 2 1 inner 672.4.k.e ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.k.e ✓ 48 1.a even 1 1 trivial
672.4.k.e ✓ 48 3.b odd 2 1 inner
672.4.k.e ✓ 48 4.b odd 2 1 inner
672.4.k.e ✓ 48 7.b odd 2 1 inner
672.4.k.e ✓ 48 12.b even 2 1 inner
672.4.k.e ✓ 48 21.c even 2 1 inner
672.4.k.e ✓ 48 28.d even 2 1 inner
672.4.k.e ✓ 48 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 870T_{5}^{10} + 258612T_{5}^{8} - 31810184T_{5}^{6} + 1487796000T_{5}^{4} - 14172053376T_{5}^{2} + 10893977600 \) acting on \(S_{4}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display