Properties

Label 1764.2.l.b
Level $1764$
Weight $2$
Character orbit 1764.l
Analytic conductor $14.086$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(949,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.l (of order \(3\), degree \(2\), not 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{3} - 2 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{3} - 2 q^{5} - 3 q^{9} + 4 q^{11} + ( - 3 \zeta_{6} + 3) q^{13} + (4 \zeta_{6} - 2) q^{15} + ( - 7 \zeta_{6} + 7) q^{17} + 5 \zeta_{6} q^{19} + 4 q^{23} - q^{25} + (6 \zeta_{6} - 3) q^{27} + \zeta_{6} q^{29} - 3 \zeta_{6} q^{31} + ( - 8 \zeta_{6} + 4) q^{33} - 11 \zeta_{6} q^{37} + ( - 3 \zeta_{6} - 3) q^{39} + (9 \zeta_{6} - 9) q^{41} - 5 \zeta_{6} q^{43} + 6 q^{45} + ( - 3 \zeta_{6} + 3) q^{47} + ( - 7 \zeta_{6} - 7) q^{51} + (3 \zeta_{6} - 3) q^{53} - 8 q^{55} + ( - 5 \zeta_{6} + 10) q^{57} - 7 \zeta_{6} q^{59} + ( - 3 \zeta_{6} + 3) q^{61} + (6 \zeta_{6} - 6) q^{65} - 13 \zeta_{6} q^{67} + ( - 8 \zeta_{6} + 4) q^{69} - 8 q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + (2 \zeta_{6} - 1) q^{75} + ( - 9 \zeta_{6} + 9) q^{79} + 9 q^{81} + \zeta_{6} q^{83} + (14 \zeta_{6} - 14) q^{85} + ( - \zeta_{6} + 2) q^{87} + 15 \zeta_{6} q^{89} + (3 \zeta_{6} - 6) q^{93} - 10 \zeta_{6} q^{95} - 17 \zeta_{6} q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 6 q^{9} + 8 q^{11} + 3 q^{13} + 7 q^{17} + 5 q^{19} + 8 q^{23} - 2 q^{25} + q^{29} - 3 q^{31} - 11 q^{37} - 9 q^{39} - 9 q^{41} - 5 q^{43} + 12 q^{45} + 3 q^{47} - 21 q^{51} - 3 q^{53} - 16 q^{55} + 15 q^{57} - 7 q^{59} + 3 q^{61} - 6 q^{65} - 13 q^{67} - 16 q^{71} + 7 q^{73} + 9 q^{79} + 18 q^{81} + q^{83} - 14 q^{85} + 3 q^{87} + 15 q^{89} - 9 q^{93} - 10 q^{95} - 17 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 −2.00000 0 0 0 −3.00000 0
961.1 0 1.73205i 0 −2.00000 0 0 0 −3.00000 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
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.l.b 2
3.b odd 2 1 5292.2.l.b 2
7.b odd 2 1 252.2.l.a yes 2
7.c even 3 1 1764.2.i.b 2
7.c even 3 1 1764.2.j.a 2
7.d odd 6 1 252.2.i.a 2
7.d odd 6 1 1764.2.j.c 2
9.c even 3 1 1764.2.i.b 2
9.d odd 6 1 5292.2.i.b 2
21.c even 2 1 756.2.l.a 2
21.g even 6 1 756.2.i.a 2
21.g even 6 1 5292.2.j.c 2
21.h odd 6 1 5292.2.i.b 2
21.h odd 6 1 5292.2.j.b 2
28.d even 2 1 1008.2.t.b 2
28.f even 6 1 1008.2.q.f 2
63.g even 3 1 inner 1764.2.l.b 2
63.h even 3 1 1764.2.j.a 2
63.i even 6 1 2268.2.k.b 2
63.i even 6 1 5292.2.j.c 2
63.j odd 6 1 5292.2.j.b 2
63.k odd 6 1 252.2.l.a yes 2
63.l odd 6 1 252.2.i.a 2
63.l odd 6 1 2268.2.k.a 2
63.n odd 6 1 5292.2.l.b 2
63.o even 6 1 756.2.i.a 2
63.o even 6 1 2268.2.k.b 2
63.s even 6 1 756.2.l.a 2
63.t odd 6 1 1764.2.j.c 2
63.t odd 6 1 2268.2.k.a 2
84.h odd 2 1 3024.2.t.b 2
84.j odd 6 1 3024.2.q.e 2
252.n even 6 1 1008.2.t.b 2
252.s odd 6 1 3024.2.q.e 2
252.bi even 6 1 1008.2.q.f 2
252.bn odd 6 1 3024.2.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 7.d odd 6 1
252.2.i.a 2 63.l odd 6 1
252.2.l.a yes 2 7.b odd 2 1
252.2.l.a yes 2 63.k odd 6 1
756.2.i.a 2 21.g even 6 1
756.2.i.a 2 63.o even 6 1
756.2.l.a 2 21.c even 2 1
756.2.l.a 2 63.s even 6 1
1008.2.q.f 2 28.f even 6 1
1008.2.q.f 2 252.bi even 6 1
1008.2.t.b 2 28.d even 2 1
1008.2.t.b 2 252.n even 6 1
1764.2.i.b 2 7.c even 3 1
1764.2.i.b 2 9.c even 3 1
1764.2.j.a 2 7.c even 3 1
1764.2.j.a 2 63.h even 3 1
1764.2.j.c 2 7.d odd 6 1
1764.2.j.c 2 63.t odd 6 1
1764.2.l.b 2 1.a even 1 1 trivial
1764.2.l.b 2 63.g even 3 1 inner
2268.2.k.a 2 63.l odd 6 1
2268.2.k.a 2 63.t odd 6 1
2268.2.k.b 2 63.i even 6 1
2268.2.k.b 2 63.o even 6 1
3024.2.q.e 2 84.j odd 6 1
3024.2.q.e 2 252.s odd 6 1
3024.2.t.b 2 84.h odd 2 1
3024.2.t.b 2 252.bn odd 6 1
5292.2.i.b 2 9.d odd 6 1
5292.2.i.b 2 21.h odd 6 1
5292.2.j.b 2 21.h odd 6 1
5292.2.j.b 2 63.j odd 6 1
5292.2.j.c 2 21.g even 6 1
5292.2.j.c 2 63.i even 6 1
5292.2.l.b 2 3.b odd 2 1
5292.2.l.b 2 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$17$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$83$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
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