Properties

Label 1008.2.bf.e
Level $1008$
Weight $2$
Character orbit 1008.bf
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
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(31,1008)]
 
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, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bf (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: \(\Q(\sqrt{2}, \sqrt{-3})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{3} + ( - \beta_{2} + 2 \beta_1 + 1) q^{5} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + (3 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{3} + ( - \beta_{2} + 2 \beta_1 + 1) q^{5} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + (3 \beta_1 + 3) q^{9} + ( - 2 \beta_1 - 1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{15} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{17} + \beta_1 q^{19} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{21} + ( - \beta_{2} + 6 \beta_1 + 3) q^{23} + (4 \beta_{3} - 2 \beta_{2} - 4) q^{25} + ( - 6 \beta_1 - 3) q^{27} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{29} - 4 \beta_1 q^{31} + 3 \beta_1 q^{33} + ( - 2 \beta_{2} + 7 \beta_1 + 5) q^{35} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{37} + ( - 2 \beta_{3} + \beta_{2} - 3) q^{39} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{41} + (2 \beta_{3} + \beta_1 + 2) q^{43} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{45} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{47} + ( - 2 \beta_{3} - 5 \beta_1) q^{49} + (4 \beta_{3} - 2 \beta_{2} - 3) q^{51} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{53} + ( - 2 \beta_{3} + \beta_{2} + 3) q^{55} + ( - \beta_1 + 1) q^{57} - 6 \beta_1 q^{59} + ( - 4 \beta_1 + 4) q^{61} + ( - 3 \beta_{3} + 3 \beta_1) q^{63} + ( - 3 \beta_1 - 3) q^{65} + ( - \beta_{3} + 2 \beta_{2} - 9 \beta_1) q^{69} + ( - 4 \beta_{2} - 4 \beta_1 - 2) q^{71} + (5 \beta_1 - 5) q^{73} + ( - 6 \beta_{3} + 4 \beta_1 + 8) q^{75} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{79} + 9 \beta_1 q^{81} + (2 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{83} + ( - 3 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 15) q^{85} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{87} + ( - 6 \beta_{3} - \beta_1 - 2) q^{89} + (2 \beta_{2} + 7 \beta_1 + 2) q^{91} + (4 \beta_1 - 4) q^{93} + (\beta_{3} - \beta_1 - 2) q^{95} + (2 \beta_{3} + \beta_1 + 2) q^{97} + ( - 3 \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 2 q^{7} + 6 q^{9} + 6 q^{13} + 6 q^{15} + 6 q^{17} - 2 q^{19} - 16 q^{25} - 6 q^{29} + 8 q^{31} - 6 q^{33} + 6 q^{35} - 2 q^{37} - 12 q^{39} + 18 q^{41} + 6 q^{43} - 18 q^{45} + 10 q^{49} - 12 q^{51} + 6 q^{53} + 12 q^{55} + 6 q^{57} + 12 q^{59} + 24 q^{61} - 6 q^{63} - 6 q^{65} + 18 q^{69} - 30 q^{73} + 24 q^{75} + 6 q^{77} - 24 q^{79} - 18 q^{81} + 6 q^{83} + 30 q^{85} + 18 q^{87} - 6 q^{89} - 6 q^{91} - 24 q^{93} - 6 q^{95} + 6 q^{97} + 18 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.707107 + 1.22474i
0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
0 −1.50000 + 0.866025i 0 4.18154i 0 2.62132 + 0.358719i 0 1.50000 2.59808i 0
31.2 0 −1.50000 + 0.866025i 0 0.717439i 0 −1.62132 2.09077i 0 1.50000 2.59808i 0
943.1 0 −1.50000 0.866025i 0 0.717439i 0 −1.62132 + 2.09077i 0 1.50000 + 2.59808i 0
943.2 0 −1.50000 0.866025i 0 4.18154i 0 2.62132 0.358719i 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
252.n 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.bf.e 4
3.b odd 2 1 3024.2.bf.f 4
4.b odd 2 1 1008.2.bf.f yes 4
7.d odd 6 1 1008.2.cz.f yes 4
9.c even 3 1 1008.2.cz.e yes 4
9.d odd 6 1 3024.2.cz.e 4
12.b even 2 1 3024.2.bf.e 4
21.g even 6 1 3024.2.cz.f 4
28.f even 6 1 1008.2.cz.e yes 4
36.f odd 6 1 1008.2.cz.f yes 4
36.h even 6 1 3024.2.cz.f 4
63.k odd 6 1 1008.2.bf.f yes 4
63.s even 6 1 3024.2.bf.e 4
84.j odd 6 1 3024.2.cz.e 4
252.n even 6 1 inner 1008.2.bf.e 4
252.bn odd 6 1 3024.2.bf.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.e 4 1.a even 1 1 trivial
1008.2.bf.e 4 252.n even 6 1 inner
1008.2.bf.f yes 4 4.b odd 2 1
1008.2.bf.f yes 4 63.k odd 6 1
1008.2.cz.e yes 4 9.c even 3 1
1008.2.cz.e yes 4 28.f even 6 1
1008.2.cz.f yes 4 7.d odd 6 1
1008.2.cz.f yes 4 36.f odd 6 1
3024.2.bf.e 4 12.b even 2 1
3024.2.bf.e 4 63.s even 6 1
3024.2.bf.f 4 3.b odd 2 1
3024.2.bf.f 4 252.bn odd 6 1
3024.2.cz.e 4 9.d odd 6 1
3024.2.cz.e 4 84.j odd 6 1
3024.2.cz.f 4 21.g even 6 1
3024.2.cz.f 4 36.h even 6 1

Hecke kernels

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

\( T_{5}^{4} + 18T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{19}^{2} + T_{19} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 18T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} - 9 T^{2} + 126 T + 441 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 66T^{2} + 441 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 45 T^{2} - 54 T + 81 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289 \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} + 111 T^{2} - 54 T + 9 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} - 9 T^{2} + 126 T + 441 \) Copy content Toggle raw display
$47$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 45 T^{2} + 54 T + 81 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 216T^{2} + 7056 \) Copy content Toggle raw display
$73$ \( (T^{2} + 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + 216 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} - 201 T^{2} + \cdots + 45369 \) Copy content Toggle raw display
$97$ \( T^{4} - 6 T^{3} - 9 T^{2} + 126 T + 441 \) Copy content Toggle raw display
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