↑

Properties

Label 1224.1.bb.a
Level $1224$
Weight $1$
Character orbit 1224.bb
Analytic conductor $0.611$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,1,Mod(67,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, 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("1224.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1224.bb (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: \(0.610855575463\)
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: \(D_{6}\)
Projective field: Galois closure of 6.2.2062988352.13

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{8} - \zeta_{6} q^{9} + (\zeta_{6} + 1) q^{11} + q^{12} + \zeta_{6}^{2} q^{16} - \zeta_{6}^{2} q^{17} - q^{18} + q^{19} + ( - \zeta_{6}^{2} + 1) q^{22} - \zeta_{6}^{2} q^{24} - \zeta_{6}^{2} q^{25} + q^{27} + \zeta_{6} q^{32} + (\zeta_{6}^{2} - 1) q^{33} - \zeta_{6} q^{34} + \zeta_{6}^{2} q^{36} - \zeta_{6}^{2} q^{38} + (\zeta_{6}^{2} - 1) q^{41} - \zeta_{6}^{2} q^{43} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{44} - \zeta_{6} q^{48} + \zeta_{6} q^{49} - \zeta_{6} q^{50} + \zeta_{6} q^{51} - \zeta_{6}^{2} q^{54} + \zeta_{6}^{2} q^{57} + \zeta_{6} q^{59} + q^{64} + (\zeta_{6}^{2} + \zeta_{6}) q^{66} - \zeta_{6} q^{67} - q^{68} + \zeta_{6} q^{72} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{73} + \zeta_{6} q^{75} - \zeta_{6} q^{76} + \zeta_{6}^{2} q^{81} + (\zeta_{6}^{2} + \zeta_{6}) q^{82} + \zeta_{6}^{2} q^{83} - \zeta_{6} q^{86} + ( - \zeta_{6} - 1) q^{88} - q^{89} - q^{96} + ( - \zeta_{6} - 1) q^{97} + q^{98} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{8} - q^{9} + 3 q^{11} + 2 q^{12} - q^{16} + q^{17} - 2 q^{18} + 2 q^{19} + 3 q^{22} + q^{24} + q^{25} + 2 q^{27} + q^{32} - 3 q^{33} - q^{34} - q^{36} + q^{38} - 3 q^{41} + q^{43} - q^{48} + q^{49} - q^{50} + q^{51} + q^{54} - q^{57} + q^{59} + 2 q^{64} - q^{67} - 2 q^{68} + q^{72} + q^{75} - q^{76} - q^{81} - 2 q^{83} - q^{86} - 3 q^{88} - 4 q^{89} - 2 q^{96} - 3 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i 0 0.500000 − 0.866025i 0 −1.00000 −0.500000 + 0.866025i 0
475.1 0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i 0 0.500000 + 0.866025i 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 \(\Q(\sqrt{-2}) \)
153.h even 6 1 inner
1224.bb odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.bb.a ✓ 2
3.b odd 2 1 3672.1.bb.a 2
8.d odd 2 1 CM 1224.1.bb.a ✓ 2
9.c even 3 1 1224.1.bb.b yes 2
9.d odd 6 1 3672.1.bb.b 2
17.b even 2 1 1224.1.bb.b yes 2
24.f even 2 1 3672.1.bb.a 2
51.c odd 2 1 3672.1.bb.b 2
72.l even 6 1 3672.1.bb.b 2
72.p odd 6 1 1224.1.bb.b yes 2
136.e odd 2 1 1224.1.bb.b yes 2
153.h even 6 1 inner 1224.1.bb.a ✓ 2
153.i odd 6 1 3672.1.bb.a 2
408.h even 2 1 3672.1.bb.b 2
1224.bb odd 6 1 inner 1224.1.bb.a ✓ 2
1224.bh even 6 1 3672.1.bb.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.bb.a ✓ 2 1.a even 1 1 trivial
1224.1.bb.a ✓ 2 8.d odd 2 1 CM
1224.1.bb.a ✓ 2 153.h even 6 1 inner
1224.1.bb.a ✓ 2 1224.bb odd 6 1 inner
1224.1.bb.b yes 2 9.c even 3 1
1224.1.bb.b yes 2 17.b even 2 1
1224.1.bb.b yes 2 72.p odd 6 1
1224.1.bb.b yes 2 136.e odd 2 1
3672.1.bb.a 2 3.b odd 2 1
3672.1.bb.a 2 24.f even 2 1
3672.1.bb.a 2 153.i odd 6 1
3672.1.bb.a 2 1224.bh even 6 1
3672.1.bb.b 2 9.d odd 6 1
3672.1.bb.b 2 51.c odd 2 1
3672.1.bb.b 2 72.l even 6 1
3672.1.bb.b 2 408.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 3T_{11} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\). 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
$11$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) 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
$41$ \( T^{2} + 3T + 3 \) 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
$67$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
show more
show less