Properties

Label 3528.1.g.b
Level $3528$
Weight $1$
Character orbit 3528.g
Analytic conductor $1.761$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -7, -168, 24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,1,Mod(883,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.883");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.g (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: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{6}, \sqrt{-7})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.5489031744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + i q^{8} + q^{16} - i q^{23} + q^{25} + i q^{29} - i q^{32} + q^{43} - 2 q^{46} - i q^{50} - i q^{53} + 2 q^{58} - q^{64} + q^{67} - i q^{71} - 2 i q^{86} + 2 i q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{16} + 2 q^{25} + 4 q^{43} - 4 q^{46} + 4 q^{58} - 2 q^{64} + 4 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
883.2 1.00000i 0 −1.00000 0 0 0 1.00000i 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 CM by \(\Q(\sqrt{-7}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
168.e odd 2 1 CM by \(\Q(\sqrt{-42}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.g.b 2
3.b odd 2 1 inner 3528.1.g.b 2
7.b odd 2 1 CM 3528.1.g.b 2
7.c even 3 2 3528.1.bx.c 4
7.d odd 6 2 3528.1.bx.c 4
8.d odd 2 1 inner 3528.1.g.b 2
21.c even 2 1 inner 3528.1.g.b 2
21.g even 6 2 3528.1.bx.c 4
21.h odd 6 2 3528.1.bx.c 4
24.f even 2 1 RM 3528.1.g.b 2
56.e even 2 1 inner 3528.1.g.b 2
56.k odd 6 2 3528.1.bx.c 4
56.m even 6 2 3528.1.bx.c 4
168.e odd 2 1 CM 3528.1.g.b 2
168.v even 6 2 3528.1.bx.c 4
168.be odd 6 2 3528.1.bx.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.g.b 2 1.a even 1 1 trivial
3528.1.g.b 2 3.b odd 2 1 inner
3528.1.g.b 2 7.b odd 2 1 CM
3528.1.g.b 2 8.d odd 2 1 inner
3528.1.g.b 2 21.c even 2 1 inner
3528.1.g.b 2 24.f even 2 1 RM
3528.1.g.b 2 56.e even 2 1 inner
3528.1.g.b 2 168.e odd 2 1 CM
3528.1.bx.c 4 7.c even 3 2
3528.1.bx.c 4 7.d odd 6 2
3528.1.bx.c 4 21.g even 6 2
3528.1.bx.c 4 21.h odd 6 2
3528.1.bx.c 4 56.k odd 6 2
3528.1.bx.c 4 56.m even 6 2
3528.1.bx.c 4 168.v even 6 2
3528.1.bx.c 4 168.be odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3528, [\chi])\):

\( T_{11} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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