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Properties

Label 3864.1.bx.e
Level $3864$
Weight $1$
Character orbit 3864.bx
Analytic conductor $1.928$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -552
Inner twists $8$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.16826668992.2

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{4} q^{2} + \zeta_{12}^{2} q^{3} - \zeta_{12}^{2} q^{4} - q^{6} + \zeta_{12}^{5} q^{7} + q^{8} + \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{4} q^{2} + \zeta_{12}^{2} q^{3} - \zeta_{12}^{2} q^{4} - q^{6} + \zeta_{12}^{5} q^{7} + q^{8} + \zeta_{12}^{4} q^{9} + (\zeta_{12}^{3} + \zeta_{12}) q^{11} - \zeta_{12}^{4} q^{12} - \zeta_{12}^{3} q^{14} + \zeta_{12}^{4} q^{16} + (\zeta_{12}^{3} + \zeta_{12}) q^{17} - \zeta_{12}^{2} q^{18} - \zeta_{12} q^{21} + (\zeta_{12}^{5} - \zeta_{12}) q^{22} - \zeta_{12}^{4} q^{23} + \zeta_{12}^{2} q^{24} - \zeta_{12}^{2} q^{25} - q^{27} + \zeta_{12} q^{28} + q^{29} - \zeta_{12}^{2} q^{32} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{33} + (\zeta_{12}^{5} - \zeta_{12}) q^{34} + q^{36} - \zeta_{12}^{5} q^{42} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{44} + \zeta_{12}^{2} q^{46} + \zeta_{12}^{4} q^{47} - q^{48} - \zeta_{12}^{4} q^{49} + q^{50} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{51} - \zeta_{12}^{4} q^{54} + \zeta_{12}^{5} q^{56} + \zeta_{12}^{4} q^{58} - \zeta_{12}^{3} q^{63} + q^{64} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{66} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{68} + q^{69} - q^{71} + \zeta_{12}^{4} q^{72} - \zeta_{12}^{2} q^{73} - \zeta_{12}^{4} q^{75} + ( - \zeta_{12}^{2} - 1) q^{77} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{79} - \zeta_{12}^{2} q^{81} + \zeta_{12}^{3} q^{84} + \zeta_{12}^{2} q^{87} + (\zeta_{12}^{3} + \zeta_{12}) q^{88} - q^{92} - \zeta_{12}^{2} q^{94} - \zeta_{12}^{4} q^{96} + \zeta_{12}^{2} q^{98} + (\zeta_{12}^{5} - \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} + 4 q^{8} - 2 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{18} + 2 q^{23} + 2 q^{24} - 2 q^{25} - 4 q^{27} + 4 q^{29} - 2 q^{32} + 4 q^{36} + 2 q^{46} - 2 q^{47} - 4 q^{48} + 2 q^{49} + 4 q^{50} + 2 q^{54} - 2 q^{58} + 4 q^{64} + 4 q^{69} - 4 q^{71} - 2 q^{72} - 2 q^{73} + 2 q^{75} - 6 q^{77} - 2 q^{81} + 2 q^{87} - 4 q^{92} - 2 q^{94} + 2 q^{96} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(967\) \(1289\) \(1933\) \(2761\) \(2857\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{12}^{4}\) \(-1\)

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
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 0 −1.00000 −0.866025 + 0.500000i 1.00000 −0.500000 + 0.866025i 0
275.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 0 −1.00000 0.866025 − 0.500000i 1.00000 −0.500000 + 0.866025i 0
3035.1 −0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 0 −1.00000 −0.866025 − 0.500000i 1.00000 −0.500000 − 0.866025i 0
3035.2 −0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 0 −1.00000 0.866025 + 0.500000i 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
23.b odd 2 1 inner
24.f even 2 1 inner
161.f odd 6 1 inner
168.v even 6 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.e ✓ 4
3.b odd 2 1 3864.1.bx.f yes 4
7.c even 3 1 inner 3864.1.bx.e ✓ 4
8.d odd 2 1 3864.1.bx.f yes 4
21.h odd 6 1 3864.1.bx.f yes 4
23.b odd 2 1 inner 3864.1.bx.e ✓ 4
24.f even 2 1 inner 3864.1.bx.e ✓ 4
56.k odd 6 1 3864.1.bx.f yes 4
69.c even 2 1 3864.1.bx.f yes 4
161.f odd 6 1 inner 3864.1.bx.e ✓ 4
168.v even 6 1 inner 3864.1.bx.e ✓ 4
184.h even 2 1 3864.1.bx.f yes 4
483.m even 6 1 3864.1.bx.f yes 4
552.h odd 2 1 CM 3864.1.bx.e ✓ 4
1288.s even 6 1 3864.1.bx.f yes 4
3864.bx odd 6 1 inner 3864.1.bx.e ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.1.bx.e ✓ 4 1.a even 1 1 trivial
3864.1.bx.e ✓ 4 7.c even 3 1 inner
3864.1.bx.e ✓ 4 23.b odd 2 1 inner
3864.1.bx.e ✓ 4 24.f even 2 1 inner
3864.1.bx.e ✓ 4 161.f odd 6 1 inner
3864.1.bx.e ✓ 4 168.v even 6 1 inner
3864.1.bx.e ✓ 4 552.h odd 2 1 CM
3864.1.bx.e ✓ 4 3864.bx odd 6 1 inner
3864.1.bx.f yes 4 3.b odd 2 1
3864.1.bx.f yes 4 8.d odd 2 1
3864.1.bx.f yes 4 21.h odd 6 1
3864.1.bx.f yes 4 56.k odd 6 1
3864.1.bx.f yes 4 69.c even 2 1
3864.1.bx.f yes 4 184.h even 2 1
3864.1.bx.f yes 4 483.m even 6 1
3864.1.bx.f yes 4 1288.s even 6 1

Hecke kernels

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

\( T_{11}^{4} + 3T_{11}^{2} + 9 \) Copy content Toggle raw display
\( T_{29} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) 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
$47$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T + 1)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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