↑

Properties

Label 1008.2.bf.h
Level $1008$
Weight $2$
Character orbit 1008.bf
Analytic conductor $8.049$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(31,1008)]
 
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, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bf (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} - 8 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{3} - 8 q^{7} - q^{9} + 3 q^{13} + 6 q^{15} - 3 q^{17} - 4 q^{19} - 20 q^{21} - 30 q^{25} + 9 q^{27} + 18 q^{29} - 17 q^{31} - 12 q^{33} - 42 q^{35} - 3 q^{37} + 42 q^{39} - 36 q^{41} - 24 q^{43} + 12 q^{45} + 21 q^{47} - 24 q^{49} - 18 q^{51} - 12 q^{53} - 30 q^{55} - 13 q^{57} + 6 q^{59} - 48 q^{61} + 6 q^{63} - 42 q^{67} - 6 q^{69} + 48 q^{73} + 33 q^{75} + 36 q^{77} + 30 q^{79} + 23 q^{81} - 48 q^{83} - 21 q^{85} + 15 q^{87} + 39 q^{89} + 9 q^{91} - 8 q^{93} + 33 q^{95} - 3 q^{97} - 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −1.72205 − 0.185824i 0 4.24307i 0 0.426811 + 2.61110i 0 2.93094 + 0.639997i 0
31.2 0 −1.55599 − 0.760852i 0 − 1.99968i 0 −2.18955 + 1.48522i 0 1.84221 + 2.36776i 0
31.3 0 −1.14980 − 1.29536i 0 2.40807i 0 0.605051 − 2.57564i 0 −0.355915 + 2.97881i 0
31.4 0 −0.823206 + 1.52392i 0 0.280665i 0 0.164674 + 2.64062i 0 −1.64466 − 2.50900i 0
31.5 0 −0.342827 − 1.69778i 0 − 3.31523i 0 −0.479859 − 2.60187i 0 −2.76494 + 1.16409i 0
31.6 0 −0.143719 + 1.72608i 0 0.784857i 0 2.01780 − 1.71128i 0 −2.95869 − 0.496139i 0
31.7 0 0.352512 − 1.69580i 0 2.86361i 0 −2.64464 + 0.0766188i 0 −2.75147 − 1.19558i 0
31.8 0 0.752143 + 1.56022i 0 − 2.81241i 0 −1.84662 − 1.89473i 0 −1.86856 + 2.34702i 0
31.9 0 1.33798 + 1.09991i 0 − 1.77292i 0 −1.23647 + 2.33905i 0 0.580394 + 2.94332i 0
31.10 0 1.36545 + 1.06563i 0 2.43186i 0 2.09792 + 1.61206i 0 0.728881 + 2.91011i 0
31.11 0 1.70675 − 0.294941i 0 − 2.85720i 0 1.73056 − 2.00129i 0 2.82602 − 1.00678i 0
31.12 0 1.72276 − 0.179167i 0 1.47736i 0 −2.64567 + 0.0201361i 0 2.93580 − 0.617323i 0
943.1 0 −1.72205 + 0.185824i 0 − 4.24307i 0 0.426811 − 2.61110i 0 2.93094 − 0.639997i 0
943.2 0 −1.55599 + 0.760852i 0 1.99968i 0 −2.18955 − 1.48522i 0 1.84221 − 2.36776i 0
943.3 0 −1.14980 + 1.29536i 0 − 2.40807i 0 0.605051 + 2.57564i 0 −0.355915 − 2.97881i 0
943.4 0 −0.823206 − 1.52392i 0 − 0.280665i 0 0.164674 − 2.64062i 0 −1.64466 + 2.50900i 0
943.5 0 −0.342827 + 1.69778i 0 3.31523i 0 −0.479859 + 2.60187i 0 −2.76494 − 1.16409i 0
943.6 0 −0.143719 − 1.72608i 0 − 0.784857i 0 2.01780 + 1.71128i 0 −2.95869 + 0.496139i 0
943.7 0 0.352512 + 1.69580i 0 − 2.86361i 0 −2.64464 − 0.0766188i 0 −2.75147 + 1.19558i 0
943.8 0 0.752143 − 1.56022i 0 2.81241i 0 −1.84662 + 1.89473i 0 −1.86856 − 2.34702i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
252.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.bf.h yes 24
3.b odd 2 1 3024.2.bf.g 24
4.b odd 2 1 1008.2.bf.g ✓ 24
7.d odd 6 1 1008.2.cz.h yes 24
9.c even 3 1 1008.2.cz.g yes 24
9.d odd 6 1 3024.2.cz.h 24
12.b even 2 1 3024.2.bf.h 24
21.g even 6 1 3024.2.cz.g 24
28.f even 6 1 1008.2.cz.g yes 24
36.f odd 6 1 1008.2.cz.h yes 24
36.h even 6 1 3024.2.cz.g 24
63.k odd 6 1 1008.2.bf.g ✓ 24
63.s even 6 1 3024.2.bf.h 24
84.j odd 6 1 3024.2.cz.h 24
252.n even 6 1 inner 1008.2.bf.h yes 24
252.bn odd 6 1 3024.2.bf.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.g ✓ 24 4.b odd 2 1
1008.2.bf.g ✓ 24 63.k odd 6 1
1008.2.bf.h yes 24 1.a even 1 1 trivial
1008.2.bf.h yes 24 252.n even 6 1 inner
1008.2.cz.g yes 24 9.c even 3 1
1008.2.cz.g yes 24 28.f even 6 1
1008.2.cz.h yes 24 7.d odd 6 1
1008.2.cz.h yes 24 36.f odd 6 1
3024.2.bf.g 24 3.b odd 2 1
3024.2.bf.g 24 252.bn odd 6 1
3024.2.bf.h 24 12.b even 2 1
3024.2.bf.h 24 63.s even 6 1
3024.2.cz.g 24 21.g even 6 1
3024.2.cz.g 24 36.h even 6 1
3024.2.cz.h 24 9.d odd 6 1
3024.2.cz.h 24 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{24} + 75 T_{5}^{22} + 2442 T_{5}^{20} + 45612 T_{5}^{18} + 542673 T_{5}^{16} + 4308822 T_{5}^{14} + \cdots + 4782969 \) Copy content Toggle raw display
\( T_{19}^{24} + 4 T_{19}^{23} + 161 T_{19}^{22} + 738 T_{19}^{21} + 16894 T_{19}^{20} + \cdots + 572426914921 \) Copy content Toggle raw display