Properties

Label 1008.4.cq.c
Level $1008$
Weight $4$
Character orbit 1008.cq
Analytic conductor $59.474$
Analytic rank $0$
Dimension $32$
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] = [1008,4,Mod(431,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.431");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.cq (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: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 36 q^{7} + 128 q^{13} + 132 q^{19} + 568 q^{25} + 684 q^{31} - 560 q^{37} + 512 q^{49} + 448 q^{61} - 1428 q^{67} - 1912 q^{73} + 1932 q^{79} - 3936 q^{85} - 4548 q^{91} - 1840 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1 0 0 0 −18.6889 + 10.7900i 0 13.4076 + 12.7764i 0 0 0
431.2 0 0 0 −15.2715 + 8.81701i 0 7.72385 16.8328i 0 0 0
431.3 0 0 0 −14.3834 + 8.30425i 0 −18.2641 3.06985i 0 0 0
431.4 0 0 0 −8.33504 + 4.81224i 0 −15.5688 10.0306i 0 0 0
431.5 0 0 0 −6.99528 + 4.03872i 0 11.5633 14.4669i 0 0 0
431.6 0 0 0 −5.40790 + 3.12225i 0 3.20776 + 18.2403i 0 0 0
431.7 0 0 0 −4.97239 + 2.87081i 0 18.5085 + 0.660448i 0 0 0
431.8 0 0 0 −1.11698 + 0.644890i 0 −11.5781 + 14.4550i 0 0 0
431.9 0 0 0 1.11698 0.644890i 0 −11.5781 + 14.4550i 0 0 0
431.10 0 0 0 4.97239 2.87081i 0 18.5085 + 0.660448i 0 0 0
431.11 0 0 0 5.40790 3.12225i 0 3.20776 + 18.2403i 0 0 0
431.12 0 0 0 6.99528 4.03872i 0 11.5633 14.4669i 0 0 0
431.13 0 0 0 8.33504 4.81224i 0 −15.5688 10.0306i 0 0 0
431.14 0 0 0 14.3834 8.30425i 0 −18.2641 3.06985i 0 0 0
431.15 0 0 0 15.2715 8.81701i 0 7.72385 16.8328i 0 0 0
431.16 0 0 0 18.6889 10.7900i 0 13.4076 + 12.7764i 0 0 0
863.1 0 0 0 −18.6889 10.7900i 0 13.4076 12.7764i 0 0 0
863.2 0 0 0 −15.2715 8.81701i 0 7.72385 + 16.8328i 0 0 0
863.3 0 0 0 −14.3834 8.30425i 0 −18.2641 + 3.06985i 0 0 0
863.4 0 0 0 −8.33504 4.81224i 0 −15.5688 + 10.0306i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.cq.c yes 32
3.b odd 2 1 inner 1008.4.cq.c yes 32
4.b odd 2 1 1008.4.cq.a 32
7.c even 3 1 1008.4.cq.a 32
12.b even 2 1 1008.4.cq.a 32
21.h odd 6 1 1008.4.cq.a 32
28.g odd 6 1 inner 1008.4.cq.c yes 32
84.n even 6 1 inner 1008.4.cq.c yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.4.cq.a 32 4.b odd 2 1
1008.4.cq.a 32 7.c even 3 1
1008.4.cq.a 32 12.b even 2 1
1008.4.cq.a 32 21.h odd 6 1
1008.4.cq.c yes 32 1.a even 1 1 trivial
1008.4.cq.c yes 32 3.b odd 2 1 inner
1008.4.cq.c yes 32 28.g odd 6 1 inner
1008.4.cq.c yes 32 84.n even 6 1 inner