Properties

Label 3780.1.de.b
Level $3780$
Weight $1$
Character orbit 3780.de
Analytic conductor $1.886$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3780,1,Mod(1979,3780)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3780, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3780.1979");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3780 = 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3780.de (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: \(1.88646574775\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1260)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.661624362000.11

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} + q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} + q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{5} q^{10} + q^{14} - \zeta_{12}^{2} q^{16} - \zeta_{12}^{4} q^{20} - \zeta_{12}^{3} q^{23} + q^{25} + \zeta_{12}^{5} q^{28} + ( - \zeta_{12}^{2} - 1) q^{29} + \zeta_{12} q^{32} - \zeta_{12} q^{35} + \zeta_{12}^{3} q^{40} - \zeta_{12}^{2} q^{41} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{43} + 2 \zeta_{12}^{2} q^{46} + (\zeta_{12}^{3} + \zeta_{12}) q^{47} + \zeta_{12}^{2} q^{49} + \zeta_{12}^{5} q^{50} - \zeta_{12}^{4} q^{56} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{58} - q^{64} + q^{70} - \zeta_{12}^{2} q^{80} + \zeta_{12} q^{82} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{83} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{86} - \zeta_{12}^{4} q^{89} - 2 \zeta_{12} q^{92} + ( - \zeta_{12}^{2} - 1) q^{94} - \zeta_{12} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 4 q^{5} + 4 q^{14} - 2 q^{16} + 2 q^{20} + 4 q^{25} - 6 q^{29} - 2 q^{41} + 4 q^{46} + 2 q^{49} + 2 q^{56} - 4 q^{64} + 4 q^{70} - 2 q^{80} + 4 q^{89} - 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3780\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\) \(1891\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{4}\) \(\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1979.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 1.00000 0 −0.866025 0.500000i 1.00000i 0 −0.866025 + 0.500000i
1979.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000 0 0.866025 + 0.500000i 1.00000i 0 0.866025 0.500000i
3419.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000 0 −0.866025 + 0.500000i 1.00000i 0 −0.866025 0.500000i
3419.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000 0 0.866025 0.500000i 1.00000i 0 0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
63.s even 6 1 inner
252.bn odd 6 1 inner
315.u even 6 1 inner
1260.de odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3780.1.de.b 4
3.b odd 2 1 1260.1.de.a yes 4
4.b odd 2 1 inner 3780.1.de.b 4
5.b even 2 1 inner 3780.1.de.b 4
7.d odd 6 1 3780.1.br.b 4
9.c even 3 1 1260.1.br.a 4
9.d odd 6 1 3780.1.br.b 4
12.b even 2 1 1260.1.de.a yes 4
15.d odd 2 1 1260.1.de.a yes 4
20.d odd 2 1 CM 3780.1.de.b 4
21.g even 6 1 1260.1.br.a 4
28.f even 6 1 3780.1.br.b 4
35.i odd 6 1 3780.1.br.b 4
36.f odd 6 1 1260.1.br.a 4
36.h even 6 1 3780.1.br.b 4
45.h odd 6 1 3780.1.br.b 4
45.j even 6 1 1260.1.br.a 4
60.h even 2 1 1260.1.de.a yes 4
63.k odd 6 1 1260.1.de.a yes 4
63.s even 6 1 inner 3780.1.de.b 4
84.j odd 6 1 1260.1.br.a 4
105.p even 6 1 1260.1.br.a 4
140.s even 6 1 3780.1.br.b 4
180.n even 6 1 3780.1.br.b 4
180.p odd 6 1 1260.1.br.a 4
252.n even 6 1 1260.1.de.a yes 4
252.bn odd 6 1 inner 3780.1.de.b 4
315.u even 6 1 inner 3780.1.de.b 4
315.bn odd 6 1 1260.1.de.a yes 4
420.be odd 6 1 1260.1.br.a 4
1260.bc even 6 1 1260.1.de.a yes 4
1260.de odd 6 1 inner 3780.1.de.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.br.a 4 9.c even 3 1
1260.1.br.a 4 21.g even 6 1
1260.1.br.a 4 36.f odd 6 1
1260.1.br.a 4 45.j even 6 1
1260.1.br.a 4 84.j odd 6 1
1260.1.br.a 4 105.p even 6 1
1260.1.br.a 4 180.p odd 6 1
1260.1.br.a 4 420.be odd 6 1
1260.1.de.a yes 4 3.b odd 2 1
1260.1.de.a yes 4 12.b even 2 1
1260.1.de.a yes 4 15.d odd 2 1
1260.1.de.a yes 4 60.h even 2 1
1260.1.de.a yes 4 63.k odd 6 1
1260.1.de.a yes 4 252.n even 6 1
1260.1.de.a yes 4 315.bn odd 6 1
1260.1.de.a yes 4 1260.bc even 6 1
3780.1.br.b 4 7.d odd 6 1
3780.1.br.b 4 9.d odd 6 1
3780.1.br.b 4 28.f even 6 1
3780.1.br.b 4 35.i odd 6 1
3780.1.br.b 4 36.h even 6 1
3780.1.br.b 4 45.h odd 6 1
3780.1.br.b 4 140.s even 6 1
3780.1.br.b 4 180.n even 6 1
3780.1.de.b 4 1.a even 1 1 trivial
3780.1.de.b 4 4.b odd 2 1 inner
3780.1.de.b 4 5.b even 2 1 inner
3780.1.de.b 4 20.d odd 2 1 CM
3780.1.de.b 4 63.s even 6 1 inner
3780.1.de.b 4 252.bn odd 6 1 inner
3780.1.de.b 4 315.u even 6 1 inner
3780.1.de.b 4 1260.de odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(3780, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) 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
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$47$ \( T^{4} + 3T^{2} + 9 \) 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
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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