Properties

Label 1260.1.bt.d
Level $1260$
Weight $1$
Character orbit 1260.bt
Analytic conductor $0.629$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
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(349,1260)]
 
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([0, 4, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.349");
 
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.bt (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 3.1.11340.2
Artin image: $C_6\times S_3$
Artin 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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{3} + \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{11} + \zeta_{6} q^{13} + q^{15} - q^{17} - \zeta_{6} q^{21} + \zeta_{6}^{2} q^{25} - q^{27} - \zeta_{6}^{2} q^{29} - \zeta_{6} q^{33} + q^{35} + 2 q^{39} - \zeta_{6}^{2} q^{45} + \zeta_{6}^{2} q^{47} - \zeta_{6} q^{49} + 2 \zeta_{6}^{2} q^{51} + q^{55} - q^{63} + 2 \zeta_{6}^{2} q^{65} - q^{71} + q^{73} + \zeta_{6} q^{75} - \zeta_{6} q^{77} - \zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} + \zeta_{6}^{2} q^{83} - 2 \zeta_{6} q^{85} - \zeta_{6} q^{87} + 2 q^{91} + \zeta_{6}^{2} q^{97} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{5} + q^{7} - q^{9} + q^{11} + 2 q^{13} + 2 q^{15} - 4 q^{17} - q^{21} - q^{25} - 2 q^{27} + q^{29} - q^{33} + 2 q^{35} + 4 q^{39} + q^{45} - q^{47} - q^{49} - 2 q^{51} + 2 q^{55} - 2 q^{63} - 2 q^{65} - 2 q^{71} + 2 q^{73} + q^{75} - q^{77} + q^{79} - q^{81} - q^{83} - 2 q^{85} - q^{87} + 4 q^{91} - q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0
769.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
9.c even 3 1 inner
315.bg 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.bt.d yes 2
3.b odd 2 1 3780.1.bt.a 2
5.b even 2 1 1260.1.bt.b 2
7.b odd 2 1 1260.1.bt.b 2
9.c even 3 1 inner 1260.1.bt.d yes 2
9.d odd 6 1 3780.1.bt.a 2
15.d odd 2 1 3780.1.bt.c 2
21.c even 2 1 3780.1.bt.c 2
35.c odd 2 1 CM 1260.1.bt.d yes 2
45.h odd 6 1 3780.1.bt.c 2
45.j even 6 1 1260.1.bt.b 2
63.l odd 6 1 1260.1.bt.b 2
63.o even 6 1 3780.1.bt.c 2
105.g even 2 1 3780.1.bt.a 2
315.z even 6 1 3780.1.bt.a 2
315.bg odd 6 1 inner 1260.1.bt.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bt.b 2 5.b even 2 1
1260.1.bt.b 2 7.b odd 2 1
1260.1.bt.b 2 45.j even 6 1
1260.1.bt.b 2 63.l odd 6 1
1260.1.bt.d yes 2 1.a even 1 1 trivial
1260.1.bt.d yes 2 9.c even 3 1 inner
1260.1.bt.d yes 2 35.c odd 2 1 CM
1260.1.bt.d yes 2 315.bg odd 6 1 inner
3780.1.bt.a 2 3.b odd 2 1
3780.1.bt.a 2 9.d odd 6 1
3780.1.bt.a 2 105.g even 2 1
3780.1.bt.a 2 315.z even 6 1
3780.1.bt.c 2 15.d odd 2 1
3780.1.bt.c 2 21.c even 2 1
3780.1.bt.c 2 45.h odd 6 1
3780.1.bt.c 2 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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