Properties

Label 1008.2.cs.q
Level $1008$
Weight $2$
Character orbit 1008.cs
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{5} + \beta_{3} q^{7} + (\beta_{3} - \beta_1) q^{11} + (4 \beta_{2} + 2) q^{13} + ( - 3 \beta_{2} - 6) q^{17} - \beta_1 q^{19} + (2 \beta_{3} + \beta_1) q^{23} + 2 \beta_{2} q^{25} + ( - \beta_{3} - \beta_1) q^{31} + (2 \beta_{3} + \beta_1) q^{35} + ( - 7 \beta_{2} - 7) q^{37} + (4 \beta_{2} + 2) q^{41} + (2 \beta_{3} + 4 \beta_1) q^{43} - 3 \beta_1 q^{47} + 7 q^{49} - 3 \beta_{2} q^{53} + 3 \beta_{3} q^{55} + (3 \beta_{3} + 3 \beta_1) q^{59} + ( - \beta_{2} + 1) q^{61} + (6 \beta_{2} + 6) q^{65} + (\beta_{3} - \beta_1) q^{67} + ( - 2 \beta_{3} - 4 \beta_1) q^{71} + ( - 3 \beta_{2} - 6) q^{73} + (7 \beta_{2} + 14) q^{77} + ( - 2 \beta_{3} - \beta_1) q^{79} - 9 q^{85} + (\beta_{2} - 1) q^{89} + ( - 2 \beta_{3} - 4 \beta_1) q^{91} + (\beta_{3} - \beta_1) q^{95} + ( - 4 \beta_{2} - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{5} - 18 q^{17} - 4 q^{25} - 14 q^{37} + 28 q^{49} + 6 q^{53} + 6 q^{61} + 12 q^{65} - 18 q^{73} + 42 q^{77} - 36 q^{85} - 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1 + \beta_{2}\) \(1\) \(1\)

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
1.32288 2.29129i
−1.32288 + 2.29129i
1.32288 + 2.29129i
−1.32288 2.29129i
0 0 0 1.50000 + 0.866025i 0 −2.64575 0 0 0
271.2 0 0 0 1.50000 + 0.866025i 0 2.64575 0 0 0
703.1 0 0 0 1.50000 0.866025i 0 −2.64575 0 0 0
703.2 0 0 0 1.50000 0.866025i 0 2.64575 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
4.b odd 2 1 inner
7.d odd 6 1 inner
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.q 4
3.b odd 2 1 112.2.p.c 4
4.b odd 2 1 inner 1008.2.cs.q 4
7.c even 3 1 7056.2.b.s 4
7.d odd 6 1 inner 1008.2.cs.q 4
7.d odd 6 1 7056.2.b.s 4
12.b even 2 1 112.2.p.c 4
21.c even 2 1 784.2.p.g 4
21.g even 6 1 112.2.p.c 4
21.g even 6 1 784.2.f.d 4
21.h odd 6 1 784.2.f.d 4
21.h odd 6 1 784.2.p.g 4
24.f even 2 1 448.2.p.c 4
24.h odd 2 1 448.2.p.c 4
28.f even 6 1 inner 1008.2.cs.q 4
28.f even 6 1 7056.2.b.s 4
28.g odd 6 1 7056.2.b.s 4
84.h odd 2 1 784.2.p.g 4
84.j odd 6 1 112.2.p.c 4
84.j odd 6 1 784.2.f.d 4
84.n even 6 1 784.2.f.d 4
84.n even 6 1 784.2.p.g 4
168.s odd 6 1 3136.2.f.f 4
168.v even 6 1 3136.2.f.f 4
168.ba even 6 1 448.2.p.c 4
168.ba even 6 1 3136.2.f.f 4
168.be odd 6 1 448.2.p.c 4
168.be odd 6 1 3136.2.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 3.b odd 2 1
112.2.p.c 4 12.b even 2 1
112.2.p.c 4 21.g even 6 1
112.2.p.c 4 84.j odd 6 1
448.2.p.c 4 24.f even 2 1
448.2.p.c 4 24.h odd 2 1
448.2.p.c 4 168.ba even 6 1
448.2.p.c 4 168.be odd 6 1
784.2.f.d 4 21.g even 6 1
784.2.f.d 4 21.h odd 6 1
784.2.f.d 4 84.j odd 6 1
784.2.f.d 4 84.n even 6 1
784.2.p.g 4 21.c even 2 1
784.2.p.g 4 21.h odd 6 1
784.2.p.g 4 84.h odd 2 1
784.2.p.g 4 84.n even 6 1
1008.2.cs.q 4 1.a even 1 1 trivial
1008.2.cs.q 4 4.b odd 2 1 inner
1008.2.cs.q 4 7.d odd 6 1 inner
1008.2.cs.q 4 28.f even 6 1 inner
3136.2.f.f 4 168.s odd 6 1
3136.2.f.f 4 168.v even 6 1
3136.2.f.f 4 168.ba even 6 1
3136.2.f.f 4 168.be odd 6 1
7056.2.b.s 4 7.c even 3 1
7056.2.b.s 4 7.d odd 6 1
7056.2.b.s 4 28.f even 6 1
7056.2.b.s 4 28.g odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 3T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{4} - 21T_{11}^{2} + 441 \) Copy content Toggle raw display
\( T_{13}^{2} + 12 \) Copy content Toggle raw display
\( T_{17}^{2} + 9T_{17} + 27 \) Copy content Toggle raw display
\( T_{19}^{4} + 7T_{19}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$23$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$37$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 84)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$53$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$61$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$71$ \( (T^{2} + 84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 21T^{2} + 441 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
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