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Properties

Label 3648.1.ba.a
Level $3648$
Weight $1$
Character orbit 3648.ba
Analytic conductor $1.821$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3648,1,Mod(863,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3648.863");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3648.ba (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: \(1.82058916609\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.11409864192.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} + \zeta_{12}^{3} q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{3} + \zeta_{12}^{3} q^{7} + \zeta_{12}^{2} q^{9} - \zeta_{12}^{2} q^{13} + \zeta_{12}^{3} q^{19} - \zeta_{12}^{4} q^{21} + \zeta_{12}^{2} q^{25} - \zeta_{12}^{3} q^{27} + (\zeta_{12}^{5} - \zeta_{12}) q^{31} + q^{37} + \zeta_{12}^{3} q^{39} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{43} - \zeta_{12}^{4} q^{57} + (\zeta_{12}^{4} - 1) q^{61} + \zeta_{12}^{5} q^{63} - \zeta_{12}^{5} q^{67} + \zeta_{12}^{4} q^{73} - \zeta_{12}^{3} q^{75} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{79} + \zeta_{12}^{4} q^{81} - \zeta_{12}^{5} q^{91} + (\zeta_{12}^{2} + 1) q^{93} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} - 2 q^{13} + 2 q^{21} + 2 q^{25} + 4 q^{37} + 2 q^{57} - 6 q^{61} - 2 q^{73} - 2 q^{81} + 6 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(1921\) \(2053\) \(2623\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.866025 − 0.500000i 0 0 0 1.00000i 0 0.500000 + 0.866025i 0
863.2 0 0.866025 + 0.500000i 0 0 0 − 1.00000i 0 0.500000 + 0.866025i 0
1247.1 0 −0.866025 + 0.500000i 0 0 0 − 1.00000i 0 0.500000 − 0.866025i 0
1247.2 0 0.866025 − 0.500000i 0 0 0 1.00000i 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner
152.l odd 6 1 inner
152.o even 6 1 inner
456.s odd 6 1 inner
456.v even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3648.1.ba.a ✓ 4
3.b odd 2 1 CM 3648.1.ba.a ✓ 4
4.b odd 2 1 inner 3648.1.ba.a ✓ 4
8.b even 2 1 3648.1.ba.b yes 4
8.d odd 2 1 3648.1.ba.b yes 4
12.b even 2 1 inner 3648.1.ba.a ✓ 4
19.d odd 6 1 3648.1.ba.b yes 4
24.f even 2 1 3648.1.ba.b yes 4
24.h odd 2 1 3648.1.ba.b yes 4
57.f even 6 1 3648.1.ba.b yes 4
76.f even 6 1 3648.1.ba.b yes 4
152.l odd 6 1 inner 3648.1.ba.a ✓ 4
152.o even 6 1 inner 3648.1.ba.a ✓ 4
228.n odd 6 1 3648.1.ba.b yes 4
456.s odd 6 1 inner 3648.1.ba.a ✓ 4
456.v even 6 1 inner 3648.1.ba.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3648.1.ba.a ✓ 4 1.a even 1 1 trivial
3648.1.ba.a ✓ 4 3.b odd 2 1 CM
3648.1.ba.a ✓ 4 4.b odd 2 1 inner
3648.1.ba.a ✓ 4 12.b even 2 1 inner
3648.1.ba.a ✓ 4 152.l odd 6 1 inner
3648.1.ba.a ✓ 4 152.o even 6 1 inner
3648.1.ba.a ✓ 4 456.s odd 6 1 inner
3648.1.ba.a ✓ 4 456.v even 6 1 inner
3648.1.ba.b yes 4 8.b even 2 1
3648.1.ba.b yes 4 8.d odd 2 1
3648.1.ba.b yes 4 19.d odd 6 1
3648.1.ba.b yes 4 24.f even 2 1
3648.1.ba.b yes 4 24.h odd 2 1
3648.1.ba.b yes 4 57.f even 6 1
3648.1.ba.b yes 4 76.f even 6 1
3648.1.ba.b yes 4 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3648, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$71$ \( T^{4} \) 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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