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Properties

Label 1008.2.q.k
Level $1008$
Weight $2$
Character orbit 1008.q
Analytic conductor $8.049$
Analytic rank $0$
Dimension $22$
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(529,1008)]
 
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, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.q (of order \(3\), 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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 2 q^{3} + 3 q^{5} + 5 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 2 q^{3} + 3 q^{5} + 5 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} + q^{19} - 2 q^{23} - 10 q^{25} + 4 q^{27} + 9 q^{29} - 8 q^{31} + 29 q^{33} - 14 q^{35} + 2 q^{37} + 16 q^{39} + 16 q^{41} + q^{45} + 10 q^{47} + 15 q^{49} - 7 q^{51} + 11 q^{53} - 22 q^{55} + 7 q^{57} - 38 q^{59} + 26 q^{61} - 48 q^{63} - 26 q^{65} + 52 q^{67} - 4 q^{69} + 48 q^{71} - 35 q^{73} + 23 q^{75} + 17 q^{77} + 20 q^{79} - 38 q^{81} + 28 q^{83} - 20 q^{85} + 33 q^{87} + 6 q^{89} + 37 q^{91} + 19 q^{93} + 24 q^{95} - 29 q^{97} + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 0 −1.73195 + 0.0184869i 0 0.790938 − 1.36994i 0 −2.57645 − 0.601597i 0 2.99932 − 0.0640368i 0
529.2 0 −1.57901 − 0.711841i 0 −1.92048 + 3.32636i 0 2.55336 − 0.693065i 0 1.98656 + 2.24801i 0
529.3 0 −1.34477 − 1.09160i 0 0.918286 − 1.59052i 0 0.361656 + 2.62092i 0 0.616838 + 2.93590i 0
529.4 0 −1.22753 + 1.22195i 0 1.76479 − 3.05671i 0 2.63986 − 0.176417i 0 0.0136831 − 2.99997i 0
529.5 0 −1.12528 + 1.31671i 0 −0.927957 + 1.60727i 0 −0.900017 − 2.48796i 0 −0.467471 − 2.96335i 0
529.6 0 −0.273951 + 1.71025i 0 −1.33401 + 2.31057i 0 −0.581213 + 2.58112i 0 −2.84990 − 0.937047i 0
529.7 0 0.455209 − 1.67116i 0 0.240694 − 0.416893i 0 1.92765 + 1.81223i 0 −2.58557 − 1.52146i 0
529.8 0 1.13364 + 1.30952i 0 −0.170100 + 0.294622i 0 2.63360 − 0.253251i 0 −0.429705 + 2.96907i 0
529.9 0 1.47982 + 0.900079i 0 1.26145 − 2.18490i 0 −2.63136 + 0.275550i 0 1.37972 + 2.66390i 0
529.10 0 1.56287 − 0.746620i 0 1.71796 − 2.97559i 0 0.727932 − 2.54364i 0 1.88512 − 2.33374i 0
529.11 0 1.65097 − 0.523731i 0 −0.841578 + 1.45766i 0 −1.65502 + 2.06419i 0 2.45141 − 1.72933i 0
625.1 0 −1.73195 − 0.0184869i 0 0.790938 + 1.36994i 0 −2.57645 + 0.601597i 0 2.99932 + 0.0640368i 0
625.2 0 −1.57901 + 0.711841i 0 −1.92048 − 3.32636i 0 2.55336 + 0.693065i 0 1.98656 − 2.24801i 0
625.3 0 −1.34477 + 1.09160i 0 0.918286 + 1.59052i 0 0.361656 − 2.62092i 0 0.616838 − 2.93590i 0
625.4 0 −1.22753 − 1.22195i 0 1.76479 + 3.05671i 0 2.63986 + 0.176417i 0 0.0136831 + 2.99997i 0
625.5 0 −1.12528 − 1.31671i 0 −0.927957 − 1.60727i 0 −0.900017 + 2.48796i 0 −0.467471 + 2.96335i 0
625.6 0 −0.273951 − 1.71025i 0 −1.33401 − 2.31057i 0 −0.581213 − 2.58112i 0 −2.84990 + 0.937047i 0
625.7 0 0.455209 + 1.67116i 0 0.240694 + 0.416893i 0 1.92765 − 1.81223i 0 −2.58557 + 1.52146i 0
625.8 0 1.13364 − 1.30952i 0 −0.170100 − 0.294622i 0 2.63360 + 0.253251i 0 −0.429705 − 2.96907i 0
625.9 0 1.47982 − 0.900079i 0 1.26145 + 2.18490i 0 −2.63136 − 0.275550i 0 1.37972 − 2.66390i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.q.k 22
3.b odd 2 1 3024.2.q.k 22
4.b odd 2 1 504.2.q.d ✓ 22
7.c even 3 1 1008.2.t.k 22
9.c even 3 1 1008.2.t.k 22
9.d odd 6 1 3024.2.t.l 22
12.b even 2 1 1512.2.q.c 22
21.h odd 6 1 3024.2.t.l 22
28.g odd 6 1 504.2.t.d yes 22
36.f odd 6 1 504.2.t.d yes 22
36.h even 6 1 1512.2.t.d 22
63.h even 3 1 inner 1008.2.q.k 22
63.j odd 6 1 3024.2.q.k 22
84.n even 6 1 1512.2.t.d 22
252.u odd 6 1 504.2.q.d ✓ 22
252.bb even 6 1 1512.2.q.c 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.d ✓ 22 4.b odd 2 1
504.2.q.d ✓ 22 252.u odd 6 1
504.2.t.d yes 22 28.g odd 6 1
504.2.t.d yes 22 36.f odd 6 1
1008.2.q.k 22 1.a even 1 1 trivial
1008.2.q.k 22 63.h even 3 1 inner
1008.2.t.k 22 7.c even 3 1
1008.2.t.k 22 9.c even 3 1
1512.2.q.c 22 12.b even 2 1
1512.2.q.c 22 252.bb even 6 1
1512.2.t.d 22 36.h even 6 1
1512.2.t.d 22 84.n even 6 1
3024.2.q.k 22 3.b odd 2 1
3024.2.q.k 22 63.j odd 6 1
3024.2.t.l 22 9.d odd 6 1
3024.2.t.l 22 21.h odd 6 1

Hecke kernels

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

\( T_{5}^{22} - 3 T_{5}^{21} + 37 T_{5}^{20} - 86 T_{5}^{19} + 790 T_{5}^{18} - 1652 T_{5}^{17} + \cdots + 217156 \) Copy content Toggle raw display
\( T_{11}^{22} - 3 T_{11}^{21} + 64 T_{11}^{20} - 165 T_{11}^{19} + 2605 T_{11}^{18} - 6138 T_{11}^{17} + \cdots + 282643344 \) Copy content Toggle raw display