Properties

Label 1008.2.bt.b
Level $1008$
Weight $2$
Character orbit 1008.bt
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(17,1008)]
 
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, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{5} + ( - 3 \beta_{2} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{5} + ( - 3 \beta_{2} + 2) q^{7} + \beta_1 q^{11} + ( - 6 \beta_{2} + 3) q^{13} + (4 \beta_{3} - 2 \beta_1) q^{17} + (\beta_{2} - 2) q^{19} + (4 \beta_{3} - 4 \beta_1) q^{23} + (\beta_{2} - 1) q^{25} - 2 \beta_{3} q^{29} + ( - \beta_{2} - 1) q^{31} + (4 \beta_{3} - 5 \beta_1) q^{35} + \beta_{2} q^{37} + (3 \beta_{3} - 6 \beta_1) q^{41} + q^{43} + ( - 5 \beta_{3} - 5 \beta_1) q^{47} + ( - 3 \beta_{2} - 5) q^{49} + 2 \beta_1 q^{53} + ( - 4 \beta_{2} + 2) q^{55} + ( - 4 \beta_{3} + 2 \beta_1) q^{59} + (2 \beta_{2} - 4) q^{61} + (9 \beta_{3} - 9 \beta_1) q^{65} + ( - 11 \beta_{2} + 11) q^{67} + 5 \beta_{3} q^{71} + (\beta_{2} + 1) q^{73} + ( - 3 \beta_{3} + 2 \beta_1) q^{77} + 5 \beta_{2} q^{79} + (3 \beta_{3} - 6 \beta_1) q^{83} + 12 q^{85} + (2 \beta_{3} + 2 \beta_1) q^{89} + ( - 3 \beta_{2} - 12) q^{91} + 3 \beta_1 q^{95} + ( - 12 \beta_{2} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{7} - 6 q^{19} - 2 q^{25} - 6 q^{31} + 2 q^{37} + 4 q^{43} - 26 q^{49} - 12 q^{61} + 22 q^{67} + 6 q^{73} + 10 q^{79} + 48 q^{85} - 54 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0 0 0 −1.22474 2.12132i 0 0.500000 2.59808i 0 0 0
17.2 0 0 0 1.22474 + 2.12132i 0 0.500000 2.59808i 0 0 0
593.1 0 0 0 −1.22474 + 2.12132i 0 0.500000 + 2.59808i 0 0 0
593.2 0 0 0 1.22474 2.12132i 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
3.b odd 2 1 inner
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 1008.2.bt.b 4
3.b odd 2 1 inner 1008.2.bt.b 4
4.b odd 2 1 63.2.p.a 4
7.c even 3 1 7056.2.k.b 4
7.d odd 6 1 inner 1008.2.bt.b 4
7.d odd 6 1 7056.2.k.b 4
12.b even 2 1 63.2.p.a 4
20.d odd 2 1 1575.2.bk.c 4
20.e even 4 2 1575.2.bc.a 8
21.g even 6 1 inner 1008.2.bt.b 4
21.g even 6 1 7056.2.k.b 4
21.h odd 6 1 7056.2.k.b 4
28.d even 2 1 441.2.p.a 4
28.f even 6 1 63.2.p.a 4
28.f even 6 1 441.2.c.a 4
28.g odd 6 1 441.2.c.a 4
28.g odd 6 1 441.2.p.a 4
36.f odd 6 1 567.2.i.d 4
36.f odd 6 1 567.2.s.d 4
36.h even 6 1 567.2.i.d 4
36.h even 6 1 567.2.s.d 4
60.h even 2 1 1575.2.bk.c 4
60.l odd 4 2 1575.2.bc.a 8
84.h odd 2 1 441.2.p.a 4
84.j odd 6 1 63.2.p.a 4
84.j odd 6 1 441.2.c.a 4
84.n even 6 1 441.2.c.a 4
84.n even 6 1 441.2.p.a 4
140.s even 6 1 1575.2.bk.c 4
140.x odd 12 2 1575.2.bc.a 8
252.n even 6 1 567.2.i.d 4
252.r odd 6 1 567.2.s.d 4
252.bj even 6 1 567.2.s.d 4
252.bn odd 6 1 567.2.i.d 4
420.be odd 6 1 1575.2.bk.c 4
420.br even 12 2 1575.2.bc.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 4.b odd 2 1
63.2.p.a 4 12.b even 2 1
63.2.p.a 4 28.f even 6 1
63.2.p.a 4 84.j odd 6 1
441.2.c.a 4 28.f even 6 1
441.2.c.a 4 28.g odd 6 1
441.2.c.a 4 84.j odd 6 1
441.2.c.a 4 84.n even 6 1
441.2.p.a 4 28.d even 2 1
441.2.p.a 4 28.g odd 6 1
441.2.p.a 4 84.h odd 2 1
441.2.p.a 4 84.n even 6 1
567.2.i.d 4 36.f odd 6 1
567.2.i.d 4 36.h even 6 1
567.2.i.d 4 252.n even 6 1
567.2.i.d 4 252.bn odd 6 1
567.2.s.d 4 36.f odd 6 1
567.2.s.d 4 36.h even 6 1
567.2.s.d 4 252.r odd 6 1
567.2.s.d 4 252.bj even 6 1
1008.2.bt.b 4 1.a even 1 1 trivial
1008.2.bt.b 4 3.b odd 2 1 inner
1008.2.bt.b 4 7.d odd 6 1 inner
1008.2.bt.b 4 21.g even 6 1 inner
1575.2.bc.a 8 20.e even 4 2
1575.2.bc.a 8 60.l odd 4 2
1575.2.bc.a 8 140.x odd 12 2
1575.2.bc.a 8 420.br even 12 2
1575.2.bk.c 4 20.d odd 2 1
1575.2.bk.c 4 60.h even 2 1
1575.2.bk.c 4 140.s even 6 1
1575.2.bk.c 4 420.be odd 6 1
7056.2.k.b 4 7.c even 3 1
7056.2.k.b 4 7.d odd 6 1
7056.2.k.b 4 21.g even 6 1
7056.2.k.b 4 21.h odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 6T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{2} + 4 \) 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
$5$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 32T^{2} + 1024 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 150 T^{2} + 22500 \) Copy content Toggle raw display
$53$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$59$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
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