Properties

Label 1323.2.f.b
Level $1323$
Weight $2$
Character orbit 1323.f
Analytic conductor $10.564$
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] = [1323,2,Mod(442,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.442");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.f (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: \(10.5642081874\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + 3 q^{8} + q^{10} + ( - 5 \zeta_{6} + 5) q^{11} - 5 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{16} + 3 q^{17} - q^{19} + (\zeta_{6} - 1) q^{20} - 5 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 5 q^{26} + (\zeta_{6} - 1) q^{29} + 5 \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{34} + 3 q^{37} + (\zeta_{6} - 1) q^{38} + 3 \zeta_{6} q^{40} + 5 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} + 5 q^{44} + 3 q^{46} - 4 \zeta_{6} q^{50} + ( - 5 \zeta_{6} + 5) q^{52} + 9 q^{53} + 5 q^{55} + \zeta_{6} q^{58} + (14 \zeta_{6} - 14) q^{61} + 7 q^{64} + ( - 5 \zeta_{6} + 5) q^{65} - 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} + 12 q^{71} - 3 q^{73} + ( - 3 \zeta_{6} + 3) q^{74} - \zeta_{6} q^{76} + (8 \zeta_{6} - 8) q^{79} + q^{80} + 5 q^{82} + ( - 9 \zeta_{6} + 9) q^{83} + 3 \zeta_{6} q^{85} - \zeta_{6} q^{86} + ( - 15 \zeta_{6} + 15) q^{88} - 13 q^{89} + (3 \zeta_{6} - 3) q^{92} - \zeta_{6} q^{95} + (9 \zeta_{6} - 9) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} + q^{5} + 6 q^{8} + 2 q^{10} + 5 q^{11} - 5 q^{13} + q^{16} + 6 q^{17} - 2 q^{19} - q^{20} - 5 q^{22} + 3 q^{23} + 4 q^{25} - 10 q^{26} - q^{29} + 5 q^{32} + 3 q^{34} + 6 q^{37} - q^{38} + 3 q^{40} + 5 q^{41} + q^{43} + 10 q^{44} + 6 q^{46} - 4 q^{50} + 5 q^{52} + 18 q^{53} + 10 q^{55} + q^{58} - 14 q^{61} + 14 q^{64} + 5 q^{65} - 4 q^{67} + 3 q^{68} + 24 q^{71} - 6 q^{73} + 3 q^{74} - q^{76} - 8 q^{79} + 2 q^{80} + 10 q^{82} + 9 q^{83} + 3 q^{85} - q^{86} + 15 q^{88} - 26 q^{89} - 3 q^{92} - q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
442.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 0.500000 0.866025i 0 0 3.00000 0 1.00000
883.1 0.500000 0.866025i 0 0.500000 + 0.866025i 0.500000 + 0.866025i 0 0 3.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.f.b 2
3.b odd 2 1 441.2.f.a 2
7.b odd 2 1 1323.2.f.a 2
7.c even 3 1 1323.2.g.a 2
7.c even 3 1 1323.2.h.a 2
7.d odd 6 1 189.2.g.a 2
7.d odd 6 1 189.2.h.a 2
9.c even 3 1 inner 1323.2.f.b 2
9.c even 3 1 3969.2.a.a 1
9.d odd 6 1 441.2.f.a 2
9.d odd 6 1 3969.2.a.f 1
21.c even 2 1 441.2.f.b 2
21.g even 6 1 63.2.g.a 2
21.g even 6 1 63.2.h.a yes 2
21.h odd 6 1 441.2.g.a 2
21.h odd 6 1 441.2.h.a 2
28.f even 6 1 3024.2.q.b 2
28.f even 6 1 3024.2.t.d 2
63.g even 3 1 1323.2.h.a 2
63.h even 3 1 1323.2.g.a 2
63.i even 6 1 63.2.g.a 2
63.i even 6 1 567.2.e.a 2
63.j odd 6 1 441.2.g.a 2
63.k odd 6 1 189.2.h.a 2
63.k odd 6 1 567.2.e.b 2
63.l odd 6 1 1323.2.f.a 2
63.l odd 6 1 3969.2.a.c 1
63.n odd 6 1 441.2.h.a 2
63.o even 6 1 441.2.f.b 2
63.o even 6 1 3969.2.a.d 1
63.s even 6 1 63.2.h.a yes 2
63.s even 6 1 567.2.e.a 2
63.t odd 6 1 189.2.g.a 2
63.t odd 6 1 567.2.e.b 2
84.j odd 6 1 1008.2.q.c 2
84.j odd 6 1 1008.2.t.d 2
252.n even 6 1 3024.2.q.b 2
252.r odd 6 1 1008.2.t.d 2
252.bj even 6 1 3024.2.t.d 2
252.bn odd 6 1 1008.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 21.g even 6 1
63.2.g.a 2 63.i even 6 1
63.2.h.a yes 2 21.g even 6 1
63.2.h.a yes 2 63.s even 6 1
189.2.g.a 2 7.d odd 6 1
189.2.g.a 2 63.t odd 6 1
189.2.h.a 2 7.d odd 6 1
189.2.h.a 2 63.k odd 6 1
441.2.f.a 2 3.b odd 2 1
441.2.f.a 2 9.d odd 6 1
441.2.f.b 2 21.c even 2 1
441.2.f.b 2 63.o even 6 1
441.2.g.a 2 21.h odd 6 1
441.2.g.a 2 63.j odd 6 1
441.2.h.a 2 21.h odd 6 1
441.2.h.a 2 63.n odd 6 1
567.2.e.a 2 63.i even 6 1
567.2.e.a 2 63.s even 6 1
567.2.e.b 2 63.k odd 6 1
567.2.e.b 2 63.t odd 6 1
1008.2.q.c 2 84.j odd 6 1
1008.2.q.c 2 252.bn odd 6 1
1008.2.t.d 2 84.j odd 6 1
1008.2.t.d 2 252.r odd 6 1
1323.2.f.a 2 7.b odd 2 1
1323.2.f.a 2 63.l odd 6 1
1323.2.f.b 2 1.a even 1 1 trivial
1323.2.f.b 2 9.c even 3 1 inner
1323.2.g.a 2 7.c even 3 1
1323.2.g.a 2 63.h even 3 1
1323.2.h.a 2 7.c even 3 1
1323.2.h.a 2 63.g even 3 1
3024.2.q.b 2 28.f even 6 1
3024.2.q.b 2 252.n even 6 1
3024.2.t.d 2 28.f even 6 1
3024.2.t.d 2 252.bj even 6 1
3969.2.a.a 1 9.c even 3 1
3969.2.a.c 1 63.l odd 6 1
3969.2.a.d 1 63.o even 6 1
3969.2.a.f 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( (T + 13)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
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