Properties

Label 1008.3.f.c
Level $1008$
Weight $3$
Character orbit 1008.f
Analytic conductor $27.466$
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] = [1008,3,Mod(433,1008)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.433");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.f (of order \(2\), degree \(1\), 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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
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 \(\beta = 2\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + (\beta - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + (\beta - 5) q^{7} - 6 q^{11} + \beta q^{13} + 4 \beta q^{17} - 5 \beta q^{19} - 30 q^{23} + q^{25} + 6 q^{29} + ( - 5 \beta - 24) q^{35} + 10 q^{37} - 10 \beta q^{41} - 10 q^{43} - 4 \beta q^{47} + ( - 10 \beta + 1) q^{49} - 90 q^{53} - 6 \beta q^{55} - 5 \beta q^{59} + 5 \beta q^{61} - 24 q^{65} + 70 q^{67} + 42 q^{71} - 22 \beta q^{73} + ( - 6 \beta + 30) q^{77} - 74 q^{79} + 13 \beta q^{83} - 96 q^{85} - 30 \beta q^{89} + ( - 5 \beta - 24) q^{91} + 120 q^{95} + 16 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{7} - 12 q^{11} - 60 q^{23} + 2 q^{25} + 12 q^{29} - 48 q^{35} + 20 q^{37} - 20 q^{43} + 2 q^{49} - 180 q^{53} - 48 q^{65} + 140 q^{67} + 84 q^{71} + 60 q^{77} - 148 q^{79} - 192 q^{85} - 48 q^{91} + 240 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
2.44949i
2.44949i
0 0 0 4.89898i 0 −5.00000 4.89898i 0 0 0
433.2 0 0 0 4.89898i 0 −5.00000 + 4.89898i 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
7.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.f.c 2
3.b odd 2 1 112.3.c.b 2
4.b odd 2 1 252.3.d.c 2
7.b odd 2 1 inner 1008.3.f.c 2
12.b even 2 1 28.3.b.a 2
21.c even 2 1 112.3.c.b 2
21.g even 6 2 784.3.s.d 4
21.h odd 6 2 784.3.s.d 4
24.f even 2 1 448.3.c.d 2
24.h odd 2 1 448.3.c.c 2
28.d even 2 1 252.3.d.c 2
28.f even 6 2 1764.3.z.i 4
28.g odd 6 2 1764.3.z.i 4
60.h even 2 1 700.3.d.a 2
60.l odd 4 2 700.3.h.a 4
84.h odd 2 1 28.3.b.a 2
84.j odd 6 2 196.3.h.b 4
84.n even 6 2 196.3.h.b 4
168.e odd 2 1 448.3.c.d 2
168.i even 2 1 448.3.c.c 2
420.o odd 2 1 700.3.d.a 2
420.w even 4 2 700.3.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.b.a 2 12.b even 2 1
28.3.b.a 2 84.h odd 2 1
112.3.c.b 2 3.b odd 2 1
112.3.c.b 2 21.c even 2 1
196.3.h.b 4 84.j odd 6 2
196.3.h.b 4 84.n even 6 2
252.3.d.c 2 4.b odd 2 1
252.3.d.c 2 28.d even 2 1
448.3.c.c 2 24.h odd 2 1
448.3.c.c 2 168.i even 2 1
448.3.c.d 2 24.f even 2 1
448.3.c.d 2 168.e odd 2 1
700.3.d.a 2 60.h even 2 1
700.3.d.a 2 420.o odd 2 1
700.3.h.a 4 60.l odd 4 2
700.3.h.a 4 420.w even 4 2
784.3.s.d 4 21.g even 6 2
784.3.s.d 4 21.h odd 6 2
1008.3.f.c 2 1.a even 1 1 trivial
1008.3.f.c 2 7.b odd 2 1 inner
1764.3.z.i 4 28.f even 6 2
1764.3.z.i 4 28.g odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{2} + 24 \) Copy content Toggle raw display
\( T_{11} + 6 \) 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
$5$ \( T^{2} + 24 \) Copy content Toggle raw display
$7$ \( T^{2} + 10T + 49 \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 24 \) Copy content Toggle raw display
$17$ \( T^{2} + 384 \) Copy content Toggle raw display
$19$ \( T^{2} + 600 \) Copy content Toggle raw display
$23$ \( (T + 30)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2400 \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 384 \) Copy content Toggle raw display
$53$ \( (T + 90)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 600 \) Copy content Toggle raw display
$61$ \( T^{2} + 600 \) Copy content Toggle raw display
$67$ \( (T - 70)^{2} \) Copy content Toggle raw display
$71$ \( (T - 42)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11616 \) Copy content Toggle raw display
$79$ \( (T + 74)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4056 \) Copy content Toggle raw display
$89$ \( T^{2} + 21600 \) Copy content Toggle raw display
$97$ \( T^{2} + 6144 \) Copy content Toggle raw display
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