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Properties

Label 1224.1.cb.a
Level $1224$
Weight $1$
Character orbit 1224.cb
Analytic conductor $0.611$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
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] = [1224,1,Mod(115,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 8, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.115");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1224.cb (of order \(12\), degree \(4\), 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: \(4\)
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: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{2} q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{2} q^{6} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( - \zeta_{12}^{3} - \zeta_{12}^{2}) q^{11} + \zeta_{12}^{3} q^{12} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{3} q^{18} - \zeta_{12}^{3} q^{19} + ( - \zeta_{12}^{4} - \zeta_{12}^{3}) q^{22} + \zeta_{12}^{4} q^{24} - \zeta_{12} q^{25} + \zeta_{12}^{3} q^{27} + \zeta_{12}^{5} q^{32} + ( - \zeta_{12}^{4} - \zeta_{12}^{3}) q^{33} - q^{34} + \zeta_{12}^{4} q^{36} - \zeta_{12}^{4} q^{38} + ( - \zeta_{12} - 1) q^{41} + (\zeta_{12}^{2} + 1) q^{43} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{44} + \zeta_{12}^{5} q^{48} + \zeta_{12}^{5} q^{49} - \zeta_{12}^{2} q^{50} - q^{51} + \zeta_{12}^{4} q^{54} - \zeta_{12}^{4} q^{57} + ( - \zeta_{12}^{4} + 1) q^{59} - q^{64} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{66} + (\zeta_{12}^{3} + \zeta_{12}) q^{67} - \zeta_{12} q^{68} + \zeta_{12}^{5} q^{72} + (\zeta_{12}^{5} - \zeta_{12}^{4}) q^{73} - \zeta_{12}^{2} q^{75} - \zeta_{12}^{5} q^{76} + \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{2} - \zeta_{12}) q^{82} + (\zeta_{12}^{3} + \zeta_{12}) q^{86} + ( - \zeta_{12}^{5} + 1) q^{88} - q^{96} + ( - \zeta_{12}^{5} + 1) q^{97} - q^{98} + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{6} + 2 q^{9} - 2 q^{11} - 2 q^{16} + 2 q^{22} - 2 q^{24} + 2 q^{33} - 4 q^{34} - 2 q^{36} + 2 q^{38} - 4 q^{41} + 6 q^{43} + 2 q^{44} - 2 q^{50} - 4 q^{51} - 2 q^{54} + 2 q^{57} + 6 q^{59} - 4 q^{64} + 2 q^{66} + 2 q^{73} - 2 q^{75} - 2 q^{81} - 2 q^{82} + 4 q^{88} - 4 q^{96} + 4 q^{97} - 4 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(\zeta_{12}^{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
259.1 −0.866025 − 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 − 1.00000i 0.500000 + 0.866025i 0
931.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i 0 0.500000 − 0.866025i 0 1.00000i 0.500000 − 0.866025i 0
1075.1 0.866025 − 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i 0 0.500000 − 0.866025i 0 − 1.00000i 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.n even 12 1 inner
1224.cb odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.cb.a ✓ 4
3.b odd 2 1 3672.1.cc.b 4
8.d odd 2 1 CM 1224.1.cb.a ✓ 4
9.c even 3 1 1224.1.cb.b yes 4
9.d odd 6 1 3672.1.cc.a 4
17.c even 4 1 1224.1.cb.b yes 4
24.f even 2 1 3672.1.cc.b 4
51.f odd 4 1 3672.1.cc.a 4
72.l even 6 1 3672.1.cc.a 4
72.p odd 6 1 1224.1.cb.b yes 4
136.j odd 4 1 1224.1.cb.b yes 4
153.m odd 12 1 3672.1.cc.b 4
153.n even 12 1 inner 1224.1.cb.a ✓ 4
408.q even 4 1 3672.1.cc.a 4
1224.ca even 12 1 3672.1.cc.b 4
1224.cb odd 12 1 inner 1224.1.cb.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.cb.a ✓ 4 1.a even 1 1 trivial
1224.1.cb.a ✓ 4 8.d odd 2 1 CM
1224.1.cb.a ✓ 4 153.n even 12 1 inner
1224.1.cb.a ✓ 4 1224.cb odd 12 1 inner
1224.1.cb.b yes 4 9.c even 3 1
1224.1.cb.b yes 4 17.c even 4 1
1224.1.cb.b yes 4 72.p odd 6 1
1224.1.cb.b yes 4 136.j odd 4 1
3672.1.cc.a 4 9.d odd 6 1
3672.1.cc.a 4 51.f odd 4 1
3672.1.cc.a 4 72.l even 6 1
3672.1.cc.a 4 408.q even 4 1
3672.1.cc.b 4 3.b odd 2 1
3672.1.cc.b 4 24.f even 2 1
3672.1.cc.b 4 153.m odd 12 1
3672.1.cc.b 4 1224.ca even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 2T_{11}^{3} + 5T_{11}^{2} + 4T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
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