Properties

Label 1800.1.bk.a
Level $1800$
Weight $1$
Character orbit 1800.bk
Analytic conductor $0.898$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,1,Mod(1051,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.1051");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1800.bk (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: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.16200.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.25920000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + q^{6} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + q^{6} + q^{8} + \zeta_{6}^{2} q^{9} - \zeta_{6}^{2} q^{11} + \zeta_{6}^{2} q^{12} + \zeta_{6}^{2} q^{16} - q^{17} - \zeta_{6} q^{18} + q^{19} + \zeta_{6} q^{22} - \zeta_{6} q^{24} + q^{27} - \zeta_{6} q^{32} - q^{33} - \zeta_{6}^{2} q^{34} + q^{36} + 2 \zeta_{6}^{2} q^{38} - \zeta_{6} q^{41} - \zeta_{6}^{2} q^{43} - q^{44} + q^{48} - \zeta_{6} q^{49} + \zeta_{6} q^{51} + \zeta_{6}^{2} q^{54} - 2 \zeta_{6} q^{57} + \zeta_{6} q^{59} + q^{64} - \zeta_{6}^{2} q^{66} - \zeta_{6} q^{67} + \zeta_{6} q^{68} + \zeta_{6}^{2} q^{72} + q^{73} - 2 \zeta_{6} q^{76} - \zeta_{6} q^{81} + 2 q^{82} - \zeta_{6}^{2} q^{83} + \zeta_{6} q^{86} - \zeta_{6}^{2} q^{88} - q^{89} + \zeta_{6}^{2} q^{96} - \zeta_{6}^{2} q^{97} + q^{98} + \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} + 2 q^{8} - q^{9} + q^{11} - q^{12} - q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + q^{22} - q^{24} + 2 q^{27} - q^{32} - 2 q^{33} + q^{34} + 2 q^{36} - 2 q^{38} - 2 q^{41} + q^{43} - 2 q^{44} + 2 q^{48} - q^{49} + q^{51} - q^{54} - 2 q^{57} + q^{59} + 2 q^{64} + q^{66} - 2 q^{67} + q^{68} - q^{72} + 4 q^{73} - 2 q^{76} - q^{81} + 4 q^{82} + q^{83} + q^{86} + q^{88} - 2 q^{89} - q^{96} + q^{97} + 2 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1051.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 0 1.00000 −0.500000 + 0.866025i 0
1651.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.bk.a 2
5.b even 2 1 1800.1.bk.e yes 2
5.c odd 4 2 1800.1.ba.c 4
8.d odd 2 1 CM 1800.1.bk.a 2
9.c even 3 1 inner 1800.1.bk.a 2
40.e odd 2 1 1800.1.bk.e yes 2
40.k even 4 2 1800.1.ba.c 4
45.j even 6 1 1800.1.bk.e yes 2
45.k odd 12 2 1800.1.ba.c 4
72.p odd 6 1 inner 1800.1.bk.a 2
360.z odd 6 1 1800.1.bk.e yes 2
360.bo even 12 2 1800.1.ba.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.1.ba.c 4 5.c odd 4 2
1800.1.ba.c 4 40.k even 4 2
1800.1.ba.c 4 45.k odd 12 2
1800.1.ba.c 4 360.bo even 12 2
1800.1.bk.a 2 1.a even 1 1 trivial
1800.1.bk.a 2 8.d odd 2 1 CM
1800.1.bk.a 2 9.c even 3 1 inner
1800.1.bk.a 2 72.p odd 6 1 inner
1800.1.bk.e yes 2 5.b even 2 1
1800.1.bk.e yes 2 40.e odd 2 1
1800.1.bk.e yes 2 45.j even 6 1
1800.1.bk.e yes 2 360.z odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \) Copy content Toggle raw display
\( T_{17} + 1 \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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