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Properties

Label 1008.2.t.l
Level $1008$
Weight $2$
Character orbit 1008.t
Analytic conductor $8.049$
Analytic rank $0$
Dimension $22$
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(193,1008)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.t (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} - 2 q^{5} + q^{7} + 6 q^{11} + 7 q^{13} + q^{15} - q^{17} - 13 q^{19} + 33 q^{21} + 44 q^{25} + 2 q^{27} - 7 q^{29} - 6 q^{31} + 9 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{43} - 17 q^{47} + 29 q^{49} + 25 q^{51} + q^{53} - 2 q^{55} - 21 q^{57} + 21 q^{59} + 31 q^{61} + 7 q^{63} - 3 q^{65} + 26 q^{67} - 40 q^{69} + 32 q^{71} + 17 q^{73} + 16 q^{75} - 4 q^{77} + 16 q^{79} + 36 q^{83} + 28 q^{85} - 7 q^{87} - 2 q^{89} - 15 q^{91} - 56 q^{93} + 24 q^{95} + 19 q^{97} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1 0 −1.72252 + 0.181425i 0 2.10440 0 1.78475 − 1.95312i 0 2.93417 − 0.625017i 0
193.2 0 −1.52946 − 0.812868i 0 −3.79940 0 −2.59312 − 0.525101i 0 1.67849 + 2.48650i 0
193.3 0 −1.13766 − 1.30604i 0 3.19500 0 −2.61289 + 0.415693i 0 −0.411479 + 2.97165i 0
193.4 0 −0.677409 + 1.59409i 0 −2.66851 0 0.654882 − 2.56342i 0 −2.08224 − 2.15970i 0
193.5 0 −0.666060 − 1.59886i 0 −0.468169 0 2.39007 + 1.13471i 0 −2.11273 + 2.12988i 0
193.6 0 0.341725 + 1.69801i 0 0.526004 0 −2.43963 − 1.02383i 0 −2.76645 + 1.16050i 0
193.7 0 0.816621 − 1.52746i 0 −1.78355 0 −1.90167 + 1.83948i 0 −1.66626 − 2.49471i 0
193.8 0 1.04208 + 1.38350i 0 −0.0619693 0 1.63689 + 2.07860i 0 −0.828124 + 2.88344i 0
193.9 0 1.07968 − 1.35436i 0 3.40736 0 2.05842 + 1.66220i 0 −0.668594 − 2.92455i 0
193.10 0 1.72608 + 0.143720i 0 2.77180 0 −0.855737 − 2.50354i 0 2.95869 + 0.496145i 0
193.11 0 1.72692 − 0.133195i 0 −4.22296 0 2.37802 − 1.15974i 0 2.96452 − 0.460034i 0
961.1 0 −1.72252 − 0.181425i 0 2.10440 0 1.78475 + 1.95312i 0 2.93417 + 0.625017i 0
961.2 0 −1.52946 + 0.812868i 0 −3.79940 0 −2.59312 + 0.525101i 0 1.67849 − 2.48650i 0
961.3 0 −1.13766 + 1.30604i 0 3.19500 0 −2.61289 − 0.415693i 0 −0.411479 − 2.97165i 0
961.4 0 −0.677409 − 1.59409i 0 −2.66851 0 0.654882 + 2.56342i 0 −2.08224 + 2.15970i 0
961.5 0 −0.666060 + 1.59886i 0 −0.468169 0 2.39007 − 1.13471i 0 −2.11273 − 2.12988i 0
961.6 0 0.341725 − 1.69801i 0 0.526004 0 −2.43963 + 1.02383i 0 −2.76645 − 1.16050i 0
961.7 0 0.816621 + 1.52746i 0 −1.78355 0 −1.90167 − 1.83948i 0 −1.66626 + 2.49471i 0
961.8 0 1.04208 − 1.38350i 0 −0.0619693 0 1.63689 − 2.07860i 0 −0.828124 − 2.88344i 0
961.9 0 1.07968 + 1.35436i 0 3.40736 0 2.05842 − 1.66220i 0 −0.668594 + 2.92455i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.11
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{11} + T_{5}^{10} - 38 T_{5}^{9} - 21 T_{5}^{8} + 513 T_{5}^{7} + 108 T_{5}^{6} - 2841 T_{5}^{5} + \cdots - 74 \) Copy content Toggle raw display
\( T_{11}^{11} - 3 T_{11}^{10} - 71 T_{11}^{9} + 231 T_{11}^{8} + 1675 T_{11}^{7} - 6195 T_{11}^{6} + \cdots + 65124 \) Copy content Toggle raw display