Properties

Label 672.4.bl.b
Level $672$
Weight $4$
Character orbit 672.bl
Analytic conductor $39.649$
Analytic rank $0$
Dimension $48$
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(31,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("672.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.bl (of order \(6\), degree \(2\), 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: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 72 q^{3} + 20 q^{7} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 72 q^{3} + 20 q^{7} - 216 q^{9} - 12 q^{11} + 28 q^{19} + 120 q^{21} + 684 q^{25} - 1296 q^{27} + 460 q^{31} - 36 q^{33} + 568 q^{35} + 252 q^{37} + 324 q^{39} + 280 q^{47} - 184 q^{49} - 392 q^{53} + 848 q^{55} + 168 q^{57} - 964 q^{59} - 600 q^{61} + 180 q^{63} + 280 q^{65} - 660 q^{67} + 324 q^{73} - 2052 q^{75} + 1568 q^{77} - 2652 q^{79} - 1944 q^{81} + 1336 q^{83} - 1056 q^{85} - 3004 q^{91} - 1380 q^{93} - 3984 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 1.50000 2.59808i 0 −18.1004 + 10.4503i 0 1.74956 + 18.4374i 0 −4.50000 7.79423i 0
31.2 0 1.50000 2.59808i 0 −16.1208 + 9.30737i 0 −4.52171 17.9598i 0 −4.50000 7.79423i 0
31.3 0 1.50000 2.59808i 0 −14.0388 + 8.10531i 0 −18.0605 + 4.10082i 0 −4.50000 7.79423i 0
31.4 0 1.50000 2.59808i 0 −13.6079 + 7.85650i 0 12.9788 + 13.2118i 0 −4.50000 7.79423i 0
31.5 0 1.50000 2.59808i 0 −12.1968 + 7.04181i 0 18.4850 1.14200i 0 −4.50000 7.79423i 0
31.6 0 1.50000 2.59808i 0 −11.2973 + 6.52253i 0 −14.5303 11.4834i 0 −4.50000 7.79423i 0
31.7 0 1.50000 2.59808i 0 −8.26531 + 4.77198i 0 −16.7811 + 7.83539i 0 −4.50000 7.79423i 0
31.8 0 1.50000 2.59808i 0 −7.66468 + 4.42521i 0 11.0120 14.8908i 0 −4.50000 7.79423i 0
31.9 0 1.50000 2.59808i 0 −4.30657 + 2.48640i 0 −9.80887 15.7094i 0 −4.50000 7.79423i 0
31.10 0 1.50000 2.59808i 0 −1.65231 + 0.953964i 0 17.8528 4.92720i 0 −4.50000 7.79423i 0
31.11 0 1.50000 2.59808i 0 −1.40368 + 0.810413i 0 8.73076 + 16.3332i 0 −4.50000 7.79423i 0
31.12 0 1.50000 2.59808i 0 0.507989 0.293287i 0 −9.34878 + 15.9875i 0 −4.50000 7.79423i 0
31.13 0 1.50000 2.59808i 0 0.775884 0.447957i 0 −6.80305 + 17.2255i 0 −4.50000 7.79423i 0
31.14 0 1.50000 2.59808i 0 0.935302 0.539997i 0 15.2722 10.4766i 0 −4.50000 7.79423i 0
31.15 0 1.50000 2.59808i 0 3.31399 1.91333i 0 12.8548 + 13.3324i 0 −4.50000 7.79423i 0
31.16 0 1.50000 2.59808i 0 6.83580 3.94665i 0 −4.47632 17.9712i 0 −4.50000 7.79423i 0
31.17 0 1.50000 2.59808i 0 7.11284 4.10660i 0 −18.3102 2.78176i 0 −4.50000 7.79423i 0
31.18 0 1.50000 2.59808i 0 7.80105 4.50394i 0 −15.6161 9.95675i 0 −4.50000 7.79423i 0
31.19 0 1.50000 2.59808i 0 7.93810 4.58307i 0 13.2705 12.9187i 0 −4.50000 7.79423i 0
31.20 0 1.50000 2.59808i 0 8.75646 5.05555i 0 −3.07975 + 18.2624i 0 −4.50000 7.79423i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.bl.b yes 48
4.b odd 2 1 672.4.bl.a 48
7.d odd 6 1 672.4.bl.a 48
28.f even 6 1 inner 672.4.bl.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.bl.a 48 4.b odd 2 1
672.4.bl.a 48 7.d odd 6 1
672.4.bl.b yes 48 1.a even 1 1 trivial
672.4.bl.b yes 48 28.f even 6 1 inner