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Properties

Label 3528.1.l.a
Level $3528$
Weight $1$
Character orbit 3528.l
Analytic conductor $1.761$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -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(685,3528)]
 
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([0, 1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.685");
 
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.l (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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 504)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.9680832.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{2} - q^{4} + (\zeta_{12}^{5} - \zeta_{12}) q^{5} - \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{3} q^{2} - q^{4} + (\zeta_{12}^{5} - \zeta_{12}) q^{5} - \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{10} - \zeta_{12}^{3} q^{11} + q^{16} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{20} + q^{22} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{25} + \zeta_{12}^{3} q^{29} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{31} + \zeta_{12}^{3} q^{32} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{40} + \zeta_{12}^{3} q^{44} + (\zeta_{12}^{5} + \zeta_{12}^{3} + \zeta_{12}) q^{50} + \zeta_{12}^{3} q^{53} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{55} - q^{58} + (\zeta_{12}^{5} - \zeta_{12}) q^{59} + (\zeta_{12}^{5} - \zeta_{12}) q^{62} - q^{64} + q^{79} + (\zeta_{12}^{5} - \zeta_{12}) q^{80} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{83} - q^{88} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{16} + 4 q^{22} + 8 q^{25} - 4 q^{58} - 4 q^{64} + 4 q^{79} - 4 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
− 1.00000i 0 −1.00000 −1.73205 0 0 1.00000i 0 1.73205i
685.2 − 1.00000i 0 −1.00000 1.73205 0 0 1.00000i 0 − 1.73205i
685.3 1.00000i 0 −1.00000 −1.73205 0 0 − 1.00000i 0 − 1.73205i
685.4 1.00000i 0 −1.00000 1.73205 0 0 − 1.00000i 0 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
56.h odd 2 1 inner
168.i 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.l.a 4
3.b odd 2 1 inner 3528.1.l.a 4
7.b odd 2 1 inner 3528.1.l.a 4
7.c even 3 1 504.1.bw.a ✓ 4
7.c even 3 1 3528.1.bw.c 4
7.d odd 6 1 504.1.bw.a ✓ 4
7.d odd 6 1 3528.1.bw.c 4
8.b even 2 1 inner 3528.1.l.a 4
21.c even 2 1 inner 3528.1.l.a 4
21.g even 6 1 504.1.bw.a ✓ 4
21.g even 6 1 3528.1.bw.c 4
21.h odd 6 1 504.1.bw.a ✓ 4
21.h odd 6 1 3528.1.bw.c 4
24.h odd 2 1 CM 3528.1.l.a 4
28.f even 6 1 2016.1.ce.a 4
28.g odd 6 1 2016.1.ce.a 4
56.h odd 2 1 inner 3528.1.l.a 4
56.j odd 6 1 504.1.bw.a ✓ 4
56.j odd 6 1 3528.1.bw.c 4
56.k odd 6 1 2016.1.ce.a 4
56.m even 6 1 2016.1.ce.a 4
56.p even 6 1 504.1.bw.a ✓ 4
56.p even 6 1 3528.1.bw.c 4
84.j odd 6 1 2016.1.ce.a 4
84.n even 6 1 2016.1.ce.a 4
168.i even 2 1 inner 3528.1.l.a 4
168.s odd 6 1 504.1.bw.a ✓ 4
168.s odd 6 1 3528.1.bw.c 4
168.v even 6 1 2016.1.ce.a 4
168.ba even 6 1 504.1.bw.a ✓ 4
168.ba even 6 1 3528.1.bw.c 4
168.be odd 6 1 2016.1.ce.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bw.a ✓ 4 7.c even 3 1
504.1.bw.a ✓ 4 7.d odd 6 1
504.1.bw.a ✓ 4 21.g even 6 1
504.1.bw.a ✓ 4 21.h odd 6 1
504.1.bw.a ✓ 4 56.j odd 6 1
504.1.bw.a ✓ 4 56.p even 6 1
504.1.bw.a ✓ 4 168.s odd 6 1
504.1.bw.a ✓ 4 168.ba even 6 1
2016.1.ce.a 4 28.f even 6 1
2016.1.ce.a 4 28.g odd 6 1
2016.1.ce.a 4 56.k odd 6 1
2016.1.ce.a 4 56.m even 6 1
2016.1.ce.a 4 84.j odd 6 1
2016.1.ce.a 4 84.n even 6 1
2016.1.ce.a 4 168.v even 6 1
2016.1.ce.a 4 168.be odd 6 1
3528.1.l.a 4 1.a even 1 1 trivial
3528.1.l.a 4 3.b odd 2 1 inner
3528.1.l.a 4 7.b odd 2 1 inner
3528.1.l.a 4 8.b even 2 1 inner
3528.1.l.a 4 21.c even 2 1 inner
3528.1.l.a 4 24.h odd 2 1 CM
3528.1.l.a 4 56.h odd 2 1 inner
3528.1.l.a 4 168.i even 2 1 inner
3528.1.bw.c 4 7.c even 3 1
3528.1.bw.c 4 7.d odd 6 1
3528.1.bw.c 4 21.g even 6 1
3528.1.bw.c 4 21.h odd 6 1
3528.1.bw.c 4 56.j odd 6 1
3528.1.bw.c 4 56.p even 6 1
3528.1.bw.c 4 168.s odd 6 1
3528.1.bw.c 4 168.ba even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

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