Properties

Label 1008.2.t.a
Level $1008$
Weight $2$
Character orbit 1008.t
Analytic conductor $8.049$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 2) q^{3} - q^{5} + (3 \zeta_{6} - 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 2) q^{3} - q^{5} + (3 \zeta_{6} - 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} - 3 q^{11} + (3 \zeta_{6} - 3) q^{13} + ( - \zeta_{6} + 2) q^{15} + ( - 5 \zeta_{6} + 5) q^{17} + 7 \zeta_{6} q^{19} + ( - 4 \zeta_{6} - 1) q^{21} - 5 q^{23} - 4 q^{25} + (6 \zeta_{6} - 3) q^{27} + \zeta_{6} q^{29} - 8 \zeta_{6} q^{31} + ( - 3 \zeta_{6} + 6) q^{33} + ( - 3 \zeta_{6} + 1) q^{35} - 3 \zeta_{6} q^{37} + ( - 6 \zeta_{6} + 3) q^{39} + ( - 5 \zeta_{6} + 5) q^{41} - 7 \zeta_{6} q^{43} + (3 \zeta_{6} - 3) q^{45} + ( - 8 \zeta_{6} + 8) q^{47} + (3 \zeta_{6} - 8) q^{49} + (10 \zeta_{6} - 5) q^{51} + ( - \zeta_{6} + 1) q^{53} + 3 q^{55} + ( - 7 \zeta_{6} - 7) q^{57} + (10 \zeta_{6} - 10) q^{61} + (3 \zeta_{6} + 6) q^{63} + ( - 3 \zeta_{6} + 3) q^{65} - 12 \zeta_{6} q^{67} + ( - 5 \zeta_{6} + 10) q^{69} - 12 q^{71} + ( - 5 \zeta_{6} + 5) q^{73} + ( - 4 \zeta_{6} + 8) q^{75} + ( - 9 \zeta_{6} + 3) q^{77} + (8 \zeta_{6} - 8) q^{79} - 9 \zeta_{6} q^{81} - 15 \zeta_{6} q^{83} + (5 \zeta_{6} - 5) q^{85} + ( - \zeta_{6} - 1) q^{87} + 5 \zeta_{6} q^{89} + ( - 3 \zeta_{6} - 6) q^{91} + (8 \zeta_{6} + 8) q^{93} - 7 \zeta_{6} q^{95} - 7 \zeta_{6} q^{97} + (9 \zeta_{6} - 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 2 q^{5} + q^{7} + 3 q^{9} - 6 q^{11} - 3 q^{13} + 3 q^{15} + 5 q^{17} + 7 q^{19} - 6 q^{21} - 10 q^{23} - 8 q^{25} + q^{29} - 8 q^{31} + 9 q^{33} - q^{35} - 3 q^{37} + 5 q^{41} - 7 q^{43} - 3 q^{45} + 8 q^{47} - 13 q^{49} + q^{53} + 6 q^{55} - 21 q^{57} - 10 q^{61} + 15 q^{63} + 3 q^{65} - 12 q^{67} + 15 q^{69} - 24 q^{71} + 5 q^{73} + 12 q^{75} - 3 q^{77} - 8 q^{79} - 9 q^{81} - 15 q^{83} - 5 q^{85} - 3 q^{87} + 5 q^{89} - 15 q^{91} + 24 q^{93} - 7 q^{95} - 7 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 + 0.866025i 0 −1.00000 0 0.500000 + 2.59808i 0 1.50000 2.59808i 0
961.1 0 −1.50000 0.866025i 0 −1.00000 0 0.500000 2.59808i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
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.a 2
3.b odd 2 1 3024.2.t.e 2
4.b odd 2 1 504.2.t.b yes 2
7.c even 3 1 1008.2.q.d 2
9.c even 3 1 1008.2.q.d 2
9.d odd 6 1 3024.2.q.c 2
12.b even 2 1 1512.2.t.b 2
21.h odd 6 1 3024.2.q.c 2
28.g odd 6 1 504.2.q.b 2
36.f odd 6 1 504.2.q.b 2
36.h even 6 1 1512.2.q.a 2
63.g even 3 1 inner 1008.2.t.a 2
63.n odd 6 1 3024.2.t.e 2
84.n even 6 1 1512.2.q.a 2
252.o even 6 1 1512.2.t.b 2
252.bl odd 6 1 504.2.t.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.b 2 28.g odd 6 1
504.2.q.b 2 36.f odd 6 1
504.2.t.b yes 2 4.b odd 2 1
504.2.t.b yes 2 252.bl odd 6 1
1008.2.q.d 2 7.c even 3 1
1008.2.q.d 2 9.c even 3 1
1008.2.t.a 2 1.a even 1 1 trivial
1008.2.t.a 2 63.g even 3 1 inner
1512.2.q.a 2 36.h even 6 1
1512.2.q.a 2 84.n even 6 1
1512.2.t.b 2 12.b even 2 1
1512.2.t.b 2 252.o even 6 1
3024.2.q.c 2 9.d odd 6 1
3024.2.q.c 2 21.h odd 6 1
3024.2.t.e 2 3.b odd 2 1
3024.2.t.e 2 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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( (T + 5)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$89$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$97$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
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