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Properties

Label 1008.2.cc.d
Level $1008$
Weight $2$
Character orbit 1008.cc
Analytic conductor $8.049$
Analytic rank $0$
Dimension $48$
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] = [1008,2,Mod(209,1008)]
 
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([0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cc (of order \(6\), degree \(2\), not 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 504)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9} - 8 q^{15} - 4 q^{21} - 12 q^{23} - 24 q^{25} - 36 q^{29} - 32 q^{39} - 12 q^{43} + 6 q^{49} - 24 q^{51} + 28 q^{57} + 14 q^{63} + 36 q^{65} - 60 q^{77} + 12 q^{79} - 36 q^{81} + 12 q^{91} + 16 q^{93} + 108 q^{95} - 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −1.73104 + 0.0591464i 0 −1.11364 − 1.92889i 0 −1.78316 − 1.95457i 0 2.99300 − 0.204770i 0
209.2 0 −1.69695 + 0.346907i 0 1.79123 + 3.10250i 0 2.56863 + 0.634134i 0 2.75931 − 1.17737i 0
209.3 0 −1.53463 − 0.803066i 0 −0.0977451 − 0.169300i 0 −0.463555 − 2.60483i 0 1.71017 + 2.46482i 0
209.4 0 −1.50601 + 0.855530i 0 −0.422480 − 0.731757i 0 −2.10980 + 1.59647i 0 1.53614 − 2.57688i 0
209.5 0 −1.31185 + 1.13095i 0 −0.965651 − 1.67256i 0 1.93133 + 1.80831i 0 0.441899 − 2.96728i 0
209.6 0 −1.29861 − 1.14613i 0 −1.79302 − 3.10561i 0 2.32553 + 1.26170i 0 0.372767 + 2.97675i 0
209.7 0 −1.27553 − 1.17176i 0 0.00869840 + 0.0150661i 0 0.514871 + 2.59517i 0 0.253935 + 2.98923i 0
209.8 0 −1.01807 + 1.40126i 0 1.60290 + 2.77631i 0 0.555071 − 2.58687i 0 −0.927062 − 2.85317i 0
209.9 0 −0.703080 − 1.58293i 0 1.16173 + 2.01217i 0 1.39373 − 2.24889i 0 −2.01136 + 2.22586i 0
209.10 0 −0.610344 − 1.62095i 0 1.91834 + 3.32266i 0 −2.41978 + 1.06989i 0 −2.25496 + 1.97867i 0
209.11 0 −0.233051 + 1.71630i 0 0.0868503 + 0.150429i 0 −1.72196 − 2.00869i 0 −2.89137 − 0.799970i 0
209.12 0 −0.0936255 + 1.72952i 0 1.26858 + 2.19724i 0 −0.521158 + 2.59391i 0 −2.98247 − 0.323854i 0
209.13 0 0.0936255 − 1.72952i 0 −1.26858 − 2.19724i 0 −1.98582 + 1.74829i 0 −2.98247 − 0.323854i 0
209.14 0 0.233051 − 1.71630i 0 −0.0868503 − 0.150429i 0 2.60056 + 0.486915i 0 −2.89137 − 0.799970i 0
209.15 0 0.610344 + 1.62095i 0 −1.91834 − 3.32266i 0 0.283334 + 2.63054i 0 −2.25496 + 1.97867i 0
209.16 0 0.703080 + 1.58293i 0 −1.16173 − 2.01217i 0 1.25073 − 2.33145i 0 −2.01136 + 2.22586i 0
209.17 0 1.01807 − 1.40126i 0 −1.60290 − 2.77631i 0 1.96276 − 1.77414i 0 −0.927062 − 2.85317i 0
209.18 0 1.27553 + 1.17176i 0 −0.00869840 − 0.0150661i 0 −2.50492 + 0.851694i 0 0.253935 + 2.98923i 0
209.19 0 1.29861 + 1.14613i 0 1.79302 + 3.10561i 0 −2.25543 − 1.38312i 0 0.372767 + 2.97675i 0
209.20 0 1.31185 − 1.13095i 0 0.965651 + 1.67256i 0 −2.53170 − 0.768427i 0 0.441899 − 2.96728i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.cc.d 48
3.b odd 2 1 3024.2.cc.d 48
4.b odd 2 1 504.2.bu.a ✓ 48
7.b odd 2 1 inner 1008.2.cc.d 48
9.c even 3 1 3024.2.cc.d 48
9.d odd 6 1 inner 1008.2.cc.d 48
12.b even 2 1 1512.2.bu.a 48
21.c even 2 1 3024.2.cc.d 48
28.d even 2 1 504.2.bu.a ✓ 48
36.f odd 6 1 1512.2.bu.a 48
36.f odd 6 1 4536.2.k.a 48
36.h even 6 1 504.2.bu.a ✓ 48
36.h even 6 1 4536.2.k.a 48
63.l odd 6 1 3024.2.cc.d 48
63.o even 6 1 inner 1008.2.cc.d 48
84.h odd 2 1 1512.2.bu.a 48
252.s odd 6 1 504.2.bu.a ✓ 48
252.s odd 6 1 4536.2.k.a 48
252.bi even 6 1 1512.2.bu.a 48
252.bi even 6 1 4536.2.k.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bu.a ✓ 48 4.b odd 2 1
504.2.bu.a ✓ 48 28.d even 2 1
504.2.bu.a ✓ 48 36.h even 6 1
504.2.bu.a ✓ 48 252.s odd 6 1
1008.2.cc.d 48 1.a even 1 1 trivial
1008.2.cc.d 48 7.b odd 2 1 inner
1008.2.cc.d 48 9.d odd 6 1 inner
1008.2.cc.d 48 63.o even 6 1 inner
1512.2.bu.a 48 12.b even 2 1
1512.2.bu.a 48 36.f odd 6 1
1512.2.bu.a 48 84.h odd 2 1
1512.2.bu.a 48 252.bi even 6 1
3024.2.cc.d 48 3.b odd 2 1
3024.2.cc.d 48 9.c even 3 1
3024.2.cc.d 48 21.c even 2 1
3024.2.cc.d 48 63.l odd 6 1
4536.2.k.a 48 36.f odd 6 1
4536.2.k.a 48 36.h even 6 1
4536.2.k.a 48 252.s odd 6 1
4536.2.k.a 48 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 72 T_{5}^{46} + 2973 T_{5}^{44} + 83564 T_{5}^{42} + 1771392 T_{5}^{40} + 29379510 T_{5}^{38} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\). Copy content Toggle raw display