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Properties

Label 3864.1.bx.a
Level $3864$
Weight $1$
Character orbit 3864.bx
Analytic conductor $1.928$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -552
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3864,1,Mod(275,3864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3864, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3864.275");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3864.bx (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: \(1.92838720881\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.27048.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.8241633792.4

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + \zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + q^{6} + \zeta_{6}^{2} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + \zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + q^{6} + \zeta_{6}^{2} q^{7} + q^{8} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{11} - \zeta_{6} q^{12} + q^{14} - \zeta_{6} q^{16} - \zeta_{6}^{2} q^{17} + \zeta_{6}^{2} q^{18} - \zeta_{6} q^{21} - q^{22} - \zeta_{6} q^{23} + \zeta_{6}^{2} q^{24} + \zeta_{6}^{2} q^{25} + q^{27} - \zeta_{6} q^{28} - q^{29} + \zeta_{6}^{2} q^{32} + \zeta_{6} q^{33} - q^{34} + q^{36} - \zeta_{6} q^{37} + \zeta_{6}^{2} q^{42} + \zeta_{6} q^{44} + \zeta_{6}^{2} q^{46} + \zeta_{6} q^{47} + q^{48} - \zeta_{6} q^{49} + q^{50} + \zeta_{6} q^{51} - \zeta_{6} q^{54} + \zeta_{6}^{2} q^{56} + \zeta_{6} q^{58} - \zeta_{6} q^{61} + q^{63} + q^{64} - \zeta_{6}^{2} q^{66} + \zeta_{6} q^{68} + q^{69} - q^{71} - \zeta_{6} q^{72} - \zeta_{6}^{2} q^{73} + 2 \zeta_{6}^{2} q^{74} - \zeta_{6} q^{75} + \zeta_{6} q^{77} + \zeta_{6} q^{79} + \zeta_{6}^{2} q^{81} + q^{83} + q^{84} - \zeta_{6}^{2} q^{87} - \zeta_{6}^{2} q^{88} - \zeta_{6} q^{89} + q^{92} - \zeta_{6}^{2} q^{94} - \zeta_{6} q^{96} + \zeta_{6}^{2} q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9} + q^{11} - q^{12} + 2 q^{14} - q^{16} + q^{17} - q^{18} - q^{21} - 2 q^{22} - q^{23} - q^{24} - q^{25} + 2 q^{27} - q^{28} - 2 q^{29} - q^{32} + q^{33} - 2 q^{34} + 2 q^{36} - 2 q^{37} - q^{42} + q^{44} - q^{46} + q^{47} + 2 q^{48} - q^{49} + 2 q^{50} + q^{51} - q^{54} - q^{56} + q^{58} - 2 q^{61} + 2 q^{63} + 2 q^{64} + q^{66} + q^{68} + 2 q^{69} - 2 q^{71} - q^{72} + q^{73} - 2 q^{74} - q^{75} + q^{77} + q^{79} - q^{81} + 4 q^{83} + 2 q^{84} + q^{87} + q^{88} - 2 q^{89} + 2 q^{92} + q^{94} - q^{96} - q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3864\mathbb{Z}\right)^\times\).

\(n\) \(967\) \(1289\) \(1933\) \(2761\) \(2857\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
275.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 0 1.00000 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 0
3035.1 −0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 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
552.h odd 2 1 CM by \(\Q(\sqrt{-138}) \)
7.c even 3 1 inner
3864.bx odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3864.1.bx.a ✓ 2
3.b odd 2 1 3864.1.bx.c yes 2
7.c even 3 1 inner 3864.1.bx.a ✓ 2
8.d odd 2 1 3864.1.bx.d yes 2
21.h odd 6 1 3864.1.bx.c yes 2
23.b odd 2 1 3864.1.bx.b yes 2
24.f even 2 1 3864.1.bx.b yes 2
56.k odd 6 1 3864.1.bx.d yes 2
69.c even 2 1 3864.1.bx.d yes 2
161.f odd 6 1 3864.1.bx.b yes 2
168.v even 6 1 3864.1.bx.b yes 2
184.h even 2 1 3864.1.bx.c yes 2
483.m even 6 1 3864.1.bx.d yes 2
552.h odd 2 1 CM 3864.1.bx.a ✓ 2
1288.s even 6 1 3864.1.bx.c yes 2
3864.bx odd 6 1 inner 3864.1.bx.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.1.bx.a ✓ 2 1.a even 1 1 trivial
3864.1.bx.a ✓ 2 7.c even 3 1 inner
3864.1.bx.a ✓ 2 552.h odd 2 1 CM
3864.1.bx.a ✓ 2 3864.bx odd 6 1 inner
3864.1.bx.b yes 2 23.b odd 2 1
3864.1.bx.b yes 2 24.f even 2 1
3864.1.bx.b yes 2 161.f odd 6 1
3864.1.bx.b yes 2 168.v even 6 1
3864.1.bx.c yes 2 3.b odd 2 1
3864.1.bx.c yes 2 21.h odd 6 1
3864.1.bx.c yes 2 184.h even 2 1
3864.1.bx.c yes 2 1288.s even 6 1
3864.1.bx.d yes 2 8.d odd 2 1
3864.1.bx.d yes 2 56.k odd 6 1
3864.1.bx.d yes 2 69.c even 2 1
3864.1.bx.d yes 2 483.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3864, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \) Copy content Toggle raw display
\( T_{29} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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