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Properties

Label 1008.2.bh.d
Level $1008$
Weight $2$
Character orbit 1008.bh
Analytic conductor $8.049$
Analytic rank $0$
Dimension $30$
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] = [1008,2,Mod(95,1008)]
 
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, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bh (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: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{3} + 2 q^{7} - q^{9} - 6 q^{11} + 12 q^{17} - 9 q^{19} + 14 q^{21} + 24 q^{23} - 36 q^{25} - 27 q^{27} + 27 q^{29} - 6 q^{31} - 20 q^{33} + 6 q^{35} + 6 q^{37} + 15 q^{39} + 9 q^{41} + 21 q^{43} - 8 q^{45} + 12 q^{49} + 15 q^{51} - 3 q^{53} + 20 q^{57} - 3 q^{59} - 3 q^{61} - 24 q^{63} + 39 q^{67} + 10 q^{69} - 18 q^{71} + 21 q^{73} - 21 q^{75} + 36 q^{77} + 33 q^{79} - 17 q^{81} + 15 q^{83} - 3 q^{85} - 78 q^{87} + 6 q^{89} + 26 q^{91} - 3 q^{93} + 27 q^{95} - 6 q^{97} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0 −1.72102 − 0.195155i 0 − 2.71056i 0 1.81297 − 1.92695i 0 2.92383 + 0.671730i 0
95.2 0 −1.64782 + 0.533567i 0 3.05131i 0 −0.624921 − 2.57089i 0 2.43061 − 1.75844i 0
95.3 0 −1.51863 − 0.832925i 0 − 1.27176i 0 −2.64542 + 0.0416030i 0 1.61247 + 2.52981i 0
95.4 0 −1.37588 − 1.05213i 0 2.89544i 0 2.24404 + 1.40153i 0 0.786065 + 2.89519i 0
95.5 0 −0.655562 − 1.60320i 0 0.811591i 0 −0.744370 + 2.53888i 0 −2.14048 + 2.10199i 0
95.6 0 −0.506211 + 1.65643i 0 − 0.127213i 0 1.61727 − 2.09390i 0 −2.48750 − 1.67700i 0
95.7 0 −0.186210 + 1.72201i 0 1.89153i 0 −2.39135 + 1.13201i 0 −2.93065 − 0.641312i 0
95.8 0 0.400164 − 1.68519i 0 3.19576i 0 −0.383649 − 2.61779i 0 −2.67974 − 1.34870i 0
95.9 0 0.912658 + 1.47209i 0 − 3.37980i 0 −1.24453 − 2.33477i 0 −1.33411 + 2.68703i 0
95.10 0 0.977987 − 1.42952i 0 − 4.10733i 0 −2.24649 + 1.39759i 0 −1.08708 − 2.79611i 0
95.11 0 0.982941 − 1.42612i 0 − 0.146175i 0 2.52001 + 0.805940i 0 −1.06765 − 2.80359i 0
95.12 0 1.19485 + 1.25393i 0 − 2.36899i 0 0.783237 + 2.52716i 0 −0.144689 + 2.99651i 0
95.13 0 1.26345 + 1.18477i 0 4.04632i 0 2.53806 + 0.747163i 0 0.192628 + 2.99381i 0
95.14 0 1.64852 − 0.531401i 0 − 0.911520i 0 2.19821 − 1.47237i 0 2.43522 − 1.75205i 0
95.15 0 1.73076 + 0.0668142i 0 0.863447i 0 −2.43307 − 1.03932i 0 2.99107 + 0.231279i 0
191.1 0 −1.72102 + 0.195155i 0 2.71056i 0 1.81297 + 1.92695i 0 2.92383 − 0.671730i 0
191.2 0 −1.64782 − 0.533567i 0 − 3.05131i 0 −0.624921 + 2.57089i 0 2.43061 + 1.75844i 0
191.3 0 −1.51863 + 0.832925i 0 1.27176i 0 −2.64542 − 0.0416030i 0 1.61247 − 2.52981i 0
191.4 0 −1.37588 + 1.05213i 0 − 2.89544i 0 2.24404 − 1.40153i 0 0.786065 − 2.89519i 0
191.5 0 −0.655562 + 1.60320i 0 − 0.811591i 0 −0.744370 − 2.53888i 0 −2.14048 − 2.10199i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
252.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.bh.d yes 30
3.b odd 2 1 3024.2.bh.d 30
4.b odd 2 1 1008.2.bh.c ✓ 30
7.c even 3 1 1008.2.cj.d yes 30
9.c even 3 1 3024.2.cj.c 30
9.d odd 6 1 1008.2.cj.c yes 30
12.b even 2 1 3024.2.bh.c 30
21.h odd 6 1 3024.2.cj.d 30
28.g odd 6 1 1008.2.cj.c yes 30
36.f odd 6 1 3024.2.cj.d 30
36.h even 6 1 1008.2.cj.d yes 30
63.g even 3 1 3024.2.bh.c 30
63.n odd 6 1 1008.2.bh.c ✓ 30
84.n even 6 1 3024.2.cj.c 30
252.o even 6 1 inner 1008.2.bh.d yes 30
252.bl odd 6 1 3024.2.bh.d 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bh.c ✓ 30 4.b odd 2 1
1008.2.bh.c ✓ 30 63.n odd 6 1
1008.2.bh.d yes 30 1.a even 1 1 trivial
1008.2.bh.d yes 30 252.o even 6 1 inner
1008.2.cj.c yes 30 9.d odd 6 1
1008.2.cj.c yes 30 28.g odd 6 1
1008.2.cj.d yes 30 7.c even 3 1
1008.2.cj.d yes 30 36.h even 6 1
3024.2.bh.c 30 12.b even 2 1
3024.2.bh.c 30 63.g even 3 1
3024.2.bh.d 30 3.b odd 2 1
3024.2.bh.d 30 252.bl odd 6 1
3024.2.cj.c 30 9.c even 3 1
3024.2.cj.c 30 84.n even 6 1
3024.2.cj.d 30 21.h odd 6 1
3024.2.cj.d 30 36.f odd 6 1

Hecke kernels

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

\( T_{5}^{30} + 93 T_{5}^{28} + 3801 T_{5}^{26} + 89963 T_{5}^{24} + 1367487 T_{5}^{22} + 13990500 T_{5}^{20} + \cdots + 84672 \) Copy content Toggle raw display
\( T_{11}^{15} + 3 T_{11}^{14} - 84 T_{11}^{13} - 165 T_{11}^{12} + 2826 T_{11}^{11} + 2904 T_{11}^{10} + \cdots - 268164 \) Copy content Toggle raw display