Properties

Label 1008.2.cs.n
Level $1008$
Weight $2$
Character orbit 1008.cs
Analytic conductor $8.049$
Analytic rank $0$
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(271,1008)]
 
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, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (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: \(2\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 4) q^{5} + ( - 3 \zeta_{6} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 4) q^{5} + ( - 3 \zeta_{6} + 2) q^{7} + (2 \zeta_{6} + 2) q^{11} + (6 \zeta_{6} - 3) q^{13} + (4 \zeta_{6} + 4) q^{17} - 7 \zeta_{6} q^{19} + ( - 7 \zeta_{6} + 7) q^{25} + (5 \zeta_{6} - 5) q^{31} + ( - 10 \zeta_{6} + 2) q^{35} - \zeta_{6} q^{37} + ( - 12 \zeta_{6} + 6) q^{41} + (2 \zeta_{6} - 1) q^{43} - 6 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + 12 q^{55} + 18 \zeta_{6} q^{65} + (\zeta_{6} + 1) q^{67} + (4 \zeta_{6} - 2) q^{71} + (5 \zeta_{6} + 5) q^{73} + ( - 8 \zeta_{6} + 10) q^{77} + (9 \zeta_{6} - 18) q^{79} + 6 q^{83} + 24 q^{85} + (4 \zeta_{6} - 8) q^{89} + (3 \zeta_{6} + 12) q^{91} + ( - 14 \zeta_{6} - 14) q^{95} + ( - 8 \zeta_{6} + 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} + q^{7} + 6 q^{11} + 12 q^{17} - 7 q^{19} + 7 q^{25} - 5 q^{31} - 6 q^{35} - q^{37} - 6 q^{47} - 13 q^{49} + 24 q^{55} + 18 q^{65} + 3 q^{67} + 15 q^{73} + 12 q^{77} - 27 q^{79} + 12 q^{83} + 48 q^{85} - 12 q^{89} + 27 q^{91} - 42 q^{95}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 3.00000 + 1.73205i 0 0.500000 + 2.59808i 0 0 0
703.1 0 0 0 3.00000 1.73205i 0 0.500000 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
28.f 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.cs.n 2
3.b odd 2 1 336.2.bl.a 2
4.b odd 2 1 1008.2.cs.m 2
7.c even 3 1 7056.2.b.l 2
7.d odd 6 1 1008.2.cs.m 2
7.d odd 6 1 7056.2.b.a 2
12.b even 2 1 336.2.bl.e yes 2
21.c even 2 1 2352.2.bl.l 2
21.g even 6 1 336.2.bl.e yes 2
21.g even 6 1 2352.2.b.a 2
21.h odd 6 1 2352.2.b.h 2
21.h odd 6 1 2352.2.bl.f 2
24.f even 2 1 1344.2.bl.d 2
24.h odd 2 1 1344.2.bl.h 2
28.f even 6 1 inner 1008.2.cs.n 2
28.f even 6 1 7056.2.b.l 2
28.g odd 6 1 7056.2.b.a 2
84.h odd 2 1 2352.2.bl.f 2
84.j odd 6 1 336.2.bl.a 2
84.j odd 6 1 2352.2.b.h 2
84.n even 6 1 2352.2.b.a 2
84.n even 6 1 2352.2.bl.l 2
168.ba even 6 1 1344.2.bl.d 2
168.be odd 6 1 1344.2.bl.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.a 2 3.b odd 2 1
336.2.bl.a 2 84.j odd 6 1
336.2.bl.e yes 2 12.b even 2 1
336.2.bl.e yes 2 21.g even 6 1
1008.2.cs.m 2 4.b odd 2 1
1008.2.cs.m 2 7.d odd 6 1
1008.2.cs.n 2 1.a even 1 1 trivial
1008.2.cs.n 2 28.f even 6 1 inner
1344.2.bl.d 2 24.f even 2 1
1344.2.bl.d 2 168.ba even 6 1
1344.2.bl.h 2 24.h odd 2 1
1344.2.bl.h 2 168.be odd 6 1
2352.2.b.a 2 21.g even 6 1
2352.2.b.a 2 84.n even 6 1
2352.2.b.h 2 21.h odd 6 1
2352.2.b.h 2 84.j odd 6 1
2352.2.bl.f 2 21.h odd 6 1
2352.2.bl.f 2 84.h odd 2 1
2352.2.bl.l 2 21.c even 2 1
2352.2.bl.l 2 84.n even 6 1
7056.2.b.a 2 7.d odd 6 1
7056.2.b.a 2 28.g odd 6 1
7056.2.b.l 2 7.c even 3 1
7056.2.b.l 2 28.f even 6 1

Hecke kernels

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

\( T_{5}^{2} - 6T_{5} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 12 \) Copy content Toggle raw display
\( T_{13}^{2} + 27 \) Copy content Toggle raw display
\( T_{17}^{2} - 12T_{17} + 48 \) Copy content Toggle raw display
\( T_{19}^{2} + 7T_{19} + 49 \) 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
$5$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 27 \) Copy content Toggle raw display
$17$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 108 \) Copy content Toggle raw display
$43$ \( T^{2} + 3 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$79$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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