Properties

Label 1008.3.cg.d
Level $1008$
Weight $3$
Character orbit 1008.cg
Analytic conductor $27.466$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(145,1008)]
 
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, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (of order \(6\), 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: \(27.4660106475\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 8) q^{7} + (14 \zeta_{6} - 7) q^{13} + ( - 21 \zeta_{6} + 42) q^{19} + (25 \zeta_{6} - 25) q^{25} + ( - 35 \zeta_{6} - 35) q^{31} - 73 \zeta_{6} q^{37} - 61 q^{43} + ( - 39 \zeta_{6} + 55) q^{49} + (56 \zeta_{6} - 112) q^{61} + (13 \zeta_{6} - 13) q^{67} + ( - 63 \zeta_{6} - 63) q^{73} + 11 \zeta_{6} q^{79} + ( - 91 \zeta_{6} + 14) q^{91} + ( - 224 \zeta_{6} + 112) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 13 q^{7} + 63 q^{19} - 25 q^{25} - 105 q^{31} - 73 q^{37} - 122 q^{43} + 71 q^{49} - 168 q^{61} - 13 q^{67} - 189 q^{73} + 11 q^{79} - 63 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −6.50000 2.59808i 0 0 0
577.1 0 0 0 0 0 −6.50000 + 2.59808i 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cg.d 2
3.b odd 2 1 CM 1008.3.cg.d 2
4.b odd 2 1 63.3.m.b 2
7.d odd 6 1 inner 1008.3.cg.d 2
12.b even 2 1 63.3.m.b 2
21.g even 6 1 inner 1008.3.cg.d 2
28.d even 2 1 441.3.m.c 2
28.f even 6 1 63.3.m.b 2
28.f even 6 1 441.3.d.c 2
28.g odd 6 1 441.3.d.c 2
28.g odd 6 1 441.3.m.c 2
84.h odd 2 1 441.3.m.c 2
84.j odd 6 1 63.3.m.b 2
84.j odd 6 1 441.3.d.c 2
84.n even 6 1 441.3.d.c 2
84.n even 6 1 441.3.m.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.m.b 2 4.b odd 2 1
63.3.m.b 2 12.b even 2 1
63.3.m.b 2 28.f even 6 1
63.3.m.b 2 84.j odd 6 1
441.3.d.c 2 28.f even 6 1
441.3.d.c 2 28.g odd 6 1
441.3.d.c 2 84.j odd 6 1
441.3.d.c 2 84.n even 6 1
441.3.m.c 2 28.d even 2 1
441.3.m.c 2 28.g odd 6 1
441.3.m.c 2 84.h odd 2 1
441.3.m.c 2 84.n even 6 1
1008.3.cg.d 2 1.a even 1 1 trivial
1008.3.cg.d 2 3.b odd 2 1 CM
1008.3.cg.d 2 7.d odd 6 1 inner
1008.3.cg.d 2 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} + 147 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 147 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 63T + 1323 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 105T + 3675 \) Copy content Toggle raw display
$37$ \( T^{2} + 73T + 5329 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 61)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 168T + 9408 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 189T + 11907 \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 37632 \) Copy content Toggle raw display
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