Properties

Label 1260.2.di.a
Level $1260$
Weight $2$
Character orbit 1260.di
Analytic conductor $10.061$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,2,Mod(709,1260)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.709");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.di (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} + 1) q^{5} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7} - 3 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} + 1) q^{5} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7} - 3 \zeta_{12}^{2} q^{9} + 2 q^{11} + 4 \zeta_{12} q^{13} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{15} + 4 \zeta_{12} q^{17} - 2 \zeta_{12}^{2} q^{19} + ( - 4 \zeta_{12}^{2} - 1) q^{21} + (4 \zeta_{12}^{3} - 3) q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + 3 \zeta_{12}^{2} q^{29} - 6 \zeta_{12}^{2} q^{31} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{33} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} + 4) q^{35} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{37} + ( - 4 \zeta_{12}^{2} + 8) q^{39} + ( - 3 \zeta_{12}^{2} + 3) q^{41} + ( - 11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{43} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 6 \zeta_{12}) q^{45} - \zeta_{12} q^{47} + ( - 5 \zeta_{12}^{2} - 3) q^{49} + ( - 4 \zeta_{12}^{2} + 8) q^{51} + 10 \zeta_{12} q^{53} + (4 \zeta_{12}^{3} + 2) q^{55} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{57} - 10 \zeta_{12}^{2} q^{59} + ( - 2 \zeta_{12}^{2} + 2) q^{61} + (6 \zeta_{12}^{3} - 9 \zeta_{12}) q^{63} + (8 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{65} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{67} + 2 q^{71} + 4 \zeta_{12} q^{73} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12} + 4) q^{75} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}) q^{77} + ( - 10 \zeta_{12}^{2} + 10) q^{79} + (9 \zeta_{12}^{2} - 9) q^{81} + (\zeta_{12}^{3} - \zeta_{12}) q^{83} + (8 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{85} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{87} + 2 \zeta_{12}^{2} q^{89} + ( - 8 \zeta_{12}^{2} + 12) q^{91} + (6 \zeta_{12}^{3} - 12 \zeta_{12}) q^{93} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12}) q^{95} + (16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{97} - 6 \zeta_{12}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 6 q^{9} + 8 q^{11} + 12 q^{15} - 4 q^{19} - 12 q^{21} - 12 q^{25} + 6 q^{29} - 12 q^{31} + 20 q^{35} + 24 q^{39} + 6 q^{41} - 6 q^{45} - 22 q^{49} + 24 q^{51} + 8 q^{55} - 20 q^{59} + 4 q^{61} - 16 q^{65} + 8 q^{71} + 24 q^{75} + 20 q^{79} - 18 q^{81} - 16 q^{85} + 4 q^{89} + 32 q^{91} - 4 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
709.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −0.866025 1.50000i 0 1.00000 + 2.00000i 0 −0.866025 2.50000i 0 −1.50000 + 2.59808i 0
709.2 0 0.866025 + 1.50000i 0 1.00000 2.00000i 0 0.866025 + 2.50000i 0 −1.50000 + 2.59808i 0
949.1 0 −0.866025 + 1.50000i 0 1.00000 2.00000i 0 −0.866025 + 2.50000i 0 −1.50000 2.59808i 0
949.2 0 0.866025 1.50000i 0 1.00000 + 2.00000i 0 0.866025 2.50000i 0 −1.50000 2.59808i 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
5.b even 2 1 inner
63.g even 3 1 inner
315.bo even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.di.a yes 4
3.b odd 2 1 3780.2.di.a 4
5.b even 2 1 inner 1260.2.di.a yes 4
7.c even 3 1 1260.2.by.a 4
9.c even 3 1 1260.2.by.a 4
9.d odd 6 1 3780.2.by.a 4
15.d odd 2 1 3780.2.di.a 4
21.h odd 6 1 3780.2.by.a 4
35.j even 6 1 1260.2.by.a 4
45.h odd 6 1 3780.2.by.a 4
45.j even 6 1 1260.2.by.a 4
63.g even 3 1 inner 1260.2.di.a yes 4
63.n odd 6 1 3780.2.di.a 4
105.o odd 6 1 3780.2.by.a 4
315.v odd 6 1 3780.2.di.a 4
315.bo even 6 1 inner 1260.2.di.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.by.a 4 7.c even 3 1
1260.2.by.a 4 9.c even 3 1
1260.2.by.a 4 35.j even 6 1
1260.2.by.a 4 45.j even 6 1
1260.2.di.a yes 4 1.a even 1 1 trivial
1260.2.di.a yes 4 5.b even 2 1 inner
1260.2.di.a yes 4 63.g even 3 1 inner
1260.2.di.a yes 4 315.bo even 6 1 inner
3780.2.by.a 4 9.d odd 6 1
3780.2.by.a 4 21.h odd 6 1
3780.2.by.a 4 45.h odd 6 1
3780.2.by.a 4 105.o odd 6 1
3780.2.di.a 4 3.b odd 2 1
3780.2.di.a 4 15.d odd 2 1
3780.2.di.a 4 63.n odd 6 1
3780.2.di.a 4 315.v odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 2 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$47$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
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