↑

Properties

Label 3640.1.dk.a
Level $3640$
Weight $1$
Character orbit 3640.dk
Analytic conductor $1.817$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -40
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3640,1,Mod(179,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 4, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3640.179");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3640.dk (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: \(1.81659664598\)
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: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.7131795944000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{5} q^{5} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12}^{5} q^{5} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{3} q^{8} + q^{9} + q^{10} - \zeta_{12}^{3} q^{13} - q^{14} + \zeta_{12}^{4} q^{16} + \zeta_{12} q^{18} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{19} + \zeta_{12} q^{20} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{23} - \zeta_{12}^{4} q^{25} - \zeta_{12}^{4} q^{26} - \zeta_{12} q^{28} + \zeta_{12}^{5} q^{32} + \zeta_{12}^{4} q^{35} + \zeta_{12}^{2} q^{36} + \zeta_{12} q^{37} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{38} + \zeta_{12}^{2} q^{40} + (\zeta_{12}^{4} - 1) q^{41} - \zeta_{12}^{5} q^{45} + ( - \zeta_{12}^{4} + 1) q^{46} + \zeta_{12}^{5} q^{47} - \zeta_{12}^{4} q^{49} - \zeta_{12}^{5} q^{50} - \zeta_{12}^{5} q^{52} - \zeta_{12}^{2} q^{56} + \zeta_{12}^{5} q^{63} - q^{64} - \zeta_{12}^{2} q^{65} + \zeta_{12}^{5} q^{70} + \zeta_{12}^{3} q^{72} + \zeta_{12}^{2} q^{74} + (\zeta_{12}^{4} - 1) q^{76} + \zeta_{12}^{3} q^{80} + q^{81} + (\zeta_{12}^{5} - \zeta_{12}) q^{82} + ( - \zeta_{12}^{2} - 1) q^{89} + q^{90} + \zeta_{12}^{2} q^{91} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{92} - q^{94} + (\zeta_{12}^{3} + \zeta_{12}) q^{95} - \zeta_{12}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 4 q^{9} + 4 q^{10} - 4 q^{14} - 2 q^{16} + 2 q^{25} + 2 q^{26} - 2 q^{35} + 2 q^{36} + 2 q^{40} - 6 q^{41} + 6 q^{46} + 2 q^{49} - 2 q^{56} - 4 q^{64} - 2 q^{65} + 2 q^{74} - 6 q^{76} + 4 q^{81} - 6 q^{89} + 4 q^{90} + 2 q^{91} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3640\mathbb{Z}\right)^\times\).

\(n\) \(521\) \(561\) \(911\) \(1457\) \(1821\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{4}\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
−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 −0.866025 + 0.500000i 0 0.866025 − 0.500000i − 1.00000i 1.00000 1.00000
179.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.866025 − 0.500000i 0 −0.866025 + 0.500000i 1.00000i 1.00000 1.00000
3579.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.866025 − 0.500000i 0 0.866025 + 0.500000i 1.00000i 1.00000 1.00000
3579.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.866025 + 0.500000i 0 −0.866025 − 0.500000i − 1.00000i 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
8.d odd 2 1 inner
91.u even 6 1 inner
455.bc even 6 1 inner
728.dd odd 6 1 inner
3640.dk odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3640.1.dk.a ✓ 4
5.b even 2 1 inner 3640.1.dk.a ✓ 4
7.c even 3 1 3640.1.gq.b yes 4
8.d odd 2 1 inner 3640.1.dk.a ✓ 4
13.e even 6 1 3640.1.gq.b yes 4
35.j even 6 1 3640.1.gq.b yes 4
40.e odd 2 1 CM 3640.1.dk.a ✓ 4
56.k odd 6 1 3640.1.gq.b yes 4
65.l even 6 1 3640.1.gq.b yes 4
91.u even 6 1 inner 3640.1.dk.a ✓ 4
104.p odd 6 1 3640.1.gq.b yes 4
280.bi odd 6 1 3640.1.gq.b yes 4
455.bc even 6 1 inner 3640.1.dk.a ✓ 4
520.cd odd 6 1 3640.1.gq.b yes 4
728.dd odd 6 1 inner 3640.1.dk.a ✓ 4
3640.dk odd 6 1 inner 3640.1.dk.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.1.dk.a ✓ 4 1.a even 1 1 trivial
3640.1.dk.a ✓ 4 5.b even 2 1 inner
3640.1.dk.a ✓ 4 8.d odd 2 1 inner
3640.1.dk.a ✓ 4 40.e odd 2 1 CM
3640.1.dk.a ✓ 4 91.u even 6 1 inner
3640.1.dk.a ✓ 4 455.bc even 6 1 inner
3640.1.dk.a ✓ 4 728.dd odd 6 1 inner
3640.1.dk.a ✓ 4 3640.dk odd 6 1 inner
3640.1.gq.b yes 4 7.c even 3 1
3640.1.gq.b yes 4 13.e even 6 1
3640.1.gq.b yes 4 35.j even 6 1
3640.1.gq.b yes 4 56.k odd 6 1
3640.1.gq.b yes 4 65.l even 6 1
3640.1.gq.b yes 4 104.p odd 6 1
3640.1.gq.b yes 4 280.bi odd 6 1
3640.1.gq.b yes 4 520.cd odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{1}^{\mathrm{new}}(3640, [\chi])\). 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
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - T^{2} + 1 \) 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
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
show more
show less