Properties

Label 1260.1.cw.c
Level $1260$
Weight $1$
Character orbit 1260.cw
Analytic conductor $0.629$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,1,Mod(499,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.499");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1260.cw (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: \(0.628821915918\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.79380.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.31752000.2

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{6} + \zeta_{6}^{2} q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{6} + \zeta_{6}^{2} q^{7} + q^{8} - \zeta_{6} q^{9} + \zeta_{6}^{2} q^{10} + \zeta_{6}^{2} q^{12} + \zeta_{6}^{2} q^{14} - \zeta_{6} q^{15} + q^{16} - \zeta_{6} q^{18} + \zeta_{6}^{2} q^{20} - \zeta_{6} q^{21} + \zeta_{6}^{2} q^{23} + \zeta_{6}^{2} q^{24} - \zeta_{6} q^{25} + q^{27} + \zeta_{6}^{2} q^{28} - \zeta_{6}^{2} q^{29} - \zeta_{6} q^{30} + q^{32} - \zeta_{6} q^{35} - \zeta_{6} q^{36} + \zeta_{6}^{2} q^{40} + \zeta_{6} q^{41} - \zeta_{6} q^{42} - \zeta_{6}^{2} q^{43} + q^{45} + 2 \zeta_{6}^{2} q^{46} - q^{47} + \zeta_{6}^{2} q^{48} - \zeta_{6} q^{49} - \zeta_{6} q^{50} + q^{54} + \zeta_{6}^{2} q^{56} - \zeta_{6}^{2} q^{58} - \zeta_{6} q^{60} + q^{61} + q^{63} + q^{64} + q^{67} - 2 \zeta_{6} q^{69} - \zeta_{6} q^{70} - \zeta_{6} q^{72} + q^{75} + \zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{82} - \zeta_{6}^{2} q^{83} - \zeta_{6} q^{84} - \zeta_{6}^{2} q^{86} + \zeta_{6} q^{87} - \zeta_{6} q^{89} + q^{90} + 2 \zeta_{6}^{2} q^{92} - q^{94} + \zeta_{6}^{2} q^{96} - \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} - q^{7} + 2 q^{8} - q^{9} - q^{10} - q^{12} - q^{14} - q^{15} + 2 q^{16} - q^{18} - q^{20} - q^{21} - 2 q^{23} - q^{24} - q^{25} + 2 q^{27} - q^{28} + q^{29} - q^{30} + 2 q^{32} - q^{35} - q^{36} - q^{40} + q^{41} - q^{42} + q^{43} + 2 q^{45} - 2 q^{46} - 2 q^{47} - q^{48} - q^{49} - q^{50} + 2 q^{54} - q^{56} + q^{58} - q^{60} + 4 q^{61} + 2 q^{63} + 2 q^{64} + 4 q^{67} - 2 q^{69} - q^{70} - q^{72} + 2 q^{75} - q^{80} - q^{81} + q^{82} + q^{83} - q^{84} + q^{86} + q^{87} - 2 q^{89} + 2 q^{90} - 2 q^{92} - 2 q^{94} - q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 −0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
1159.1 1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
\(n\): 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 \(\Q(\sqrt{-5}) \)
63.h even 3 1 inner
1260.cw odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} + 2T_{23} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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