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Properties

Label 3744.1.ci.a
Level $3744$
Weight $1$
Character orbit 3744.ci
Analytic conductor $1.868$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -104
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,1,Mod(1039,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("3744.1039");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.ci (of order \(6\), degree \(2\), not 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: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 936)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.62171080298496.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{18}^{2} q^{3} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{5} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{7} + \zeta_{18}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18}^{2} q^{3} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{5} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{7} + \zeta_{18}^{4} q^{9} + \zeta_{18}^{6} q^{13} + ( - \zeta_{18}^{6} + \zeta_{18}) q^{15} + (\zeta_{18}^{8} - \zeta_{18}) q^{17} + ( - \zeta_{18} + 1) q^{21} + (\zeta_{18}^{8} + \cdots - \zeta_{18}^{3}) q^{25} + \cdots - \zeta_{18}^{8} q^{93} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{13} + 3 q^{15} + 6 q^{21} - 3 q^{25} + 3 q^{27} - 3 q^{31} + 6 q^{35} - 3 q^{45} - 3 q^{49} + 3 q^{51} + 3 q^{63} - 6 q^{75} + 3 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{18}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1039.1
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.173648 − 0.984808i
0.939693 − 0.342020i
−0.766044 − 0.642788i
−0.173648 + 0.984808i
0 −0.766044 − 0.642788i 0 −0.766044 + 1.32683i 0 0.173648 + 0.300767i 0 0.173648 + 0.984808i 0
1039.2 0 −0.173648 + 0.984808i 0 −0.173648 + 0.300767i 0 −0.939693 − 1.62760i 0 −0.939693 − 0.342020i 0
1039.3 0 0.939693 − 0.342020i 0 0.939693 − 1.62760i 0 0.766044 + 1.32683i 0 0.766044 − 0.642788i 0
3535.1 0 −0.766044 + 0.642788i 0 −0.766044 − 1.32683i 0 0.173648 − 0.300767i 0 0.173648 − 0.984808i 0
3535.2 0 −0.173648 − 0.984808i 0 −0.173648 − 0.300767i 0 −0.939693 + 1.62760i 0 −0.939693 + 0.342020i 0
3535.3 0 0.939693 + 0.342020i 0 0.939693 + 1.62760i 0 0.766044 − 1.32683i 0 0.766044 + 0.642788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1039.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
9.c even 3 1 inner
936.bs odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.1.ci.a 6
4.b odd 2 1 936.1.bs.a ✓ 6
8.b even 2 1 936.1.bs.b yes 6
8.d odd 2 1 3744.1.ci.b 6
9.c even 3 1 inner 3744.1.ci.a 6
12.b even 2 1 2808.1.bs.b 6
13.b even 2 1 3744.1.ci.b 6
24.h odd 2 1 2808.1.bs.a 6
36.f odd 6 1 936.1.bs.a ✓ 6
36.h even 6 1 2808.1.bs.b 6
52.b odd 2 1 936.1.bs.b yes 6
72.j odd 6 1 2808.1.bs.a 6
72.n even 6 1 936.1.bs.b yes 6
72.p odd 6 1 3744.1.ci.b 6
104.e even 2 1 936.1.bs.a ✓ 6
104.h odd 2 1 CM 3744.1.ci.a 6
117.t even 6 1 3744.1.ci.b 6
156.h even 2 1 2808.1.bs.a 6
312.b odd 2 1 2808.1.bs.b 6
468.x even 6 1 2808.1.bs.a 6
468.bg odd 6 1 936.1.bs.b yes 6
936.bs odd 6 1 inner 3744.1.ci.a 6
936.bx even 6 1 936.1.bs.a ✓ 6
936.cv odd 6 1 2808.1.bs.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.1.bs.a ✓ 6 4.b odd 2 1
936.1.bs.a ✓ 6 36.f odd 6 1
936.1.bs.a ✓ 6 104.e even 2 1
936.1.bs.a ✓ 6 936.bx even 6 1
936.1.bs.b yes 6 8.b even 2 1
936.1.bs.b yes 6 52.b odd 2 1
936.1.bs.b yes 6 72.n even 6 1
936.1.bs.b yes 6 468.bg odd 6 1
2808.1.bs.a 6 24.h odd 2 1
2808.1.bs.a 6 72.j odd 6 1
2808.1.bs.a 6 156.h even 2 1
2808.1.bs.a 6 468.x even 6 1
2808.1.bs.b 6 12.b even 2 1
2808.1.bs.b 6 36.h even 6 1
2808.1.bs.b 6 312.b odd 2 1
2808.1.bs.b 6 936.cv odd 6 1
3744.1.ci.a 6 1.a even 1 1 trivial
3744.1.ci.a 6 9.c even 3 1 inner
3744.1.ci.a 6 104.h odd 2 1 CM
3744.1.ci.a 6 936.bs odd 6 1 inner
3744.1.ci.b 6 8.d odd 2 1
3744.1.ci.b 6 13.b even 2 1
3744.1.ci.b 6 72.p odd 6 1
3744.1.ci.b 6 117.t even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 3T_{5}^{4} + 2T_{5}^{3} + 9T_{5}^{2} + 3T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3744, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$17$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$37$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( (T^{3} - 3 T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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