Properties

Label 1008.3.m.e
Level $1008$
Weight $3$
Character orbit 1008.m
Analytic conductor $27.466$
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] = [1008,3,Mod(127,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.m (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{5} - \beta_1 q^{7} + (\beta_{3} + 4 \beta_1) q^{11} + ( - \beta_{2} + 2) q^{13} + (\beta_{2} - 14) q^{17} + (\beta_{3} + 8 \beta_1) q^{19} + ( - 3 \beta_{3} + 4 \beta_1) q^{23} - 21 q^{25} + (2 \beta_{2} + 14) q^{29} - 4 \beta_{3} q^{31} - 2 \beta_1 q^{35} + ( - 2 \beta_{2} + 26) q^{37} + ( - \beta_{2} + 10) q^{41} + (4 \beta_{3} + 8 \beta_1) q^{43} + (4 \beta_{3} + 24 \beta_1) q^{47} - 7 q^{49} + (2 \beta_{2} - 10) q^{53} + (2 \beta_{3} + 8 \beta_1) q^{55} + (3 \beta_{3} + 8 \beta_1) q^{59} + ( - \beta_{2} - 6) q^{61} + ( - 2 \beta_{2} + 4) q^{65} - 8 \beta_{3} q^{67} + ( - 3 \beta_{3} + 28 \beta_1) q^{71} + (2 \beta_{2} + 58) q^{73} + ( - \beta_{2} + 28) q^{77} + 14 \beta_{3} q^{79} + ( - 3 \beta_{3} + 32 \beta_1) q^{83} + (2 \beta_{2} - 28) q^{85} + ( - 3 \beta_{2} - 38) q^{89} + ( - 7 \beta_{3} - 2 \beta_1) q^{91} + (2 \beta_{3} + 16 \beta_1) q^{95} + (4 \beta_{2} - 22) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} + 8 q^{13} - 56 q^{17} - 84 q^{25} + 56 q^{29} + 104 q^{37} + 40 q^{41} - 28 q^{49} - 40 q^{53} - 24 q^{61} + 16 q^{65} + 232 q^{73} + 112 q^{77} - 112 q^{85} - 152 q^{89} - 88 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\nu^{3} + 4\nu^{2} + 12\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\nu^{3} - 4\nu^{2} + 4\nu + 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 4\beta _1 + 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{3} + \beta_{2} - 4\beta _1 + 12 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta _1 + 5 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.39564 + 0.228425i
−0.895644 + 1.09445i
−0.895644 1.09445i
1.39564 0.228425i
0 0 0 2.00000 0 2.64575i 0 0 0
127.2 0 0 0 2.00000 0 2.64575i 0 0 0
127.3 0 0 0 2.00000 0 2.64575i 0 0 0
127.4 0 0 0 2.00000 0 2.64575i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.m.e 4
3.b odd 2 1 336.3.m.b 4
4.b odd 2 1 inner 1008.3.m.e 4
12.b even 2 1 336.3.m.b 4
21.c even 2 1 2352.3.m.i 4
24.f even 2 1 1344.3.m.b 4
24.h odd 2 1 1344.3.m.b 4
84.h odd 2 1 2352.3.m.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.m.b 4 3.b odd 2 1
336.3.m.b 4 12.b even 2 1
1008.3.m.e 4 1.a even 1 1 trivial
1008.3.m.e 4 4.b odd 2 1 inner
1344.3.m.b 4 24.f even 2 1
1344.3.m.b 4 24.h odd 2 1
2352.3.m.i 4 21.c even 2 1
2352.3.m.i 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2 \) acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\). 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
$5$ \( (T - 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 320T^{2} + 4096 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 332)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 28 T - 140)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 992 T^{2} + 160000 \) Copy content Toggle raw display
$23$ \( T^{4} + 1088 T^{2} + 102400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 28 T - 1148)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 768)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 52 T - 668)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 20 T - 236)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 2432 T^{2} + 102400 \) Copy content Toggle raw display
$47$ \( T^{4} + 9600 T^{2} + \cdots + 10653696 \) Copy content Toggle raw display
$53$ \( (T^{2} + 20 T - 1244)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 1760T^{2} + 256 \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T - 300)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 3072)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 11840 T^{2} + \cdots + 25563136 \) Copy content Toggle raw display
$73$ \( (T^{2} - 116 T + 2020)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 9408)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 15200 T^{2} + \cdots + 45373696 \) Copy content Toggle raw display
$89$ \( (T^{2} + 76 T - 1580)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 44 T - 4892)^{2} \) Copy content Toggle raw display
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