Properties

Label 1287.2.a.j
Level $1287$
Weight $2$
Character orbit 1287.a
Self dual yes
Analytic conductor $10.277$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1287,2,Mod(1,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1287.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.a (trivial)

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: yes
Analytic conductor: \(10.2767467401\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_1 + 3) q^{10} - q^{11} - q^{13} + (\beta_{2} + 2 \beta_1 + 1) q^{14} + (2 \beta_1 + 1) q^{16} + ( - \beta_1 + 1) q^{17} + ( - \beta_{2} + 3) q^{19} + (\beta_{2} + \beta_1 - 4) q^{20} - \beta_1 q^{22} + ( - \beta_{2} + 2 \beta_1 + 3) q^{23} + ( - 3 \beta_{2} + 2) q^{25} - \beta_1 q^{26} + (\beta_{2} + 2 \beta_1 + 7) q^{28} + ( - 2 \beta_{2} + \beta_1 - 5) q^{29} + (\beta_{2} - \beta_1 + 6) q^{31} + ( - \beta_1 + 6) q^{32} + ( - \beta_{2} + \beta_1 - 4) q^{34} + (2 \beta_{2} - 4) q^{35} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{37} + ( - \beta_{2} + 2 \beta_1 - 1) q^{38} + (2 \beta_{2} - \beta_1 - 1) q^{40} + ( - \beta_{2} - 4 \beta_1 + 1) q^{41} + (2 \beta_{2} + \beta_1 - 5) q^{43} + ( - \beta_{2} - 2) q^{44} + (\beta_{2} + 2 \beta_1 + 7) q^{46} + (2 \beta_{2} - 2 \beta_1 + 2) q^{47} + (2 \beta_1 - 1) q^{49} + ( - 3 \beta_{2} - \beta_1 - 3) q^{50} + ( - \beta_{2} - 2) q^{52} + ( - 2 \beta_1 + 4) q^{53} + (\beta_{2} - \beta_1) q^{55} + (\beta_{2} + 4 \beta_1 + 7) q^{56} + ( - \beta_{2} - 7 \beta_1 + 2) q^{58} + (2 \beta_{2} - 2 \beta_1 - 4) q^{59} + (2 \beta_1 + 2) q^{61} + (7 \beta_1 - 3) q^{62} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + (\beta_{2} - \beta_1) q^{65} + ( - \beta_{2} + 3 \beta_1 + 8) q^{67} + ( - 3 \beta_1 + 1) q^{68} + (2 \beta_{2} - 2 \beta_1 + 2) q^{70} - 2 \beta_{2} q^{71} + ( - 3 \beta_{2} - 4 \beta_1 + 3) q^{73} + (2 \beta_{2} - 4 \beta_1 + 14) q^{74} + (3 \beta_{2} - 2 \beta_1 + 1) q^{76} + ( - \beta_{2} - 1) q^{77} + ( - \beta_1 + 7) q^{79} + ( - \beta_{2} - \beta_1 + 6) q^{80} + ( - 5 \beta_{2} - 17) q^{82} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{83} + ( - \beta_{2} + 2 \beta_1 - 3) q^{85} + (3 \beta_{2} - 3 \beta_1 + 6) q^{86} + ( - \beta_{2} - \beta_1 - 1) q^{88} + (3 \beta_{2} - \beta_1 + 8) q^{89} + ( - \beta_{2} - 1) q^{91} + (5 \beta_{2} + 4 \beta_1 + 3) q^{92} + (4 \beta_1 - 6) q^{94} + ( - 6 \beta_{2} + 4 \beta_1 + 4) q^{95} + (2 \beta_{2} - 4 \beta_1 - 2) q^{97} + (2 \beta_{2} - \beta_1 + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} + 2 q^{5} + 2 q^{7} + 3 q^{8} + 8 q^{10} - 3 q^{11} - 3 q^{13} + 4 q^{14} + 5 q^{16} + 2 q^{17} + 10 q^{19} - 12 q^{20} - q^{22} + 12 q^{23} + 9 q^{25} - q^{26} + 22 q^{28} - 12 q^{29} + 16 q^{31} + 17 q^{32} - 10 q^{34} - 14 q^{35} - 6 q^{40} - 16 q^{43} - 5 q^{44} + 22 q^{46} + 2 q^{47} - q^{49} - 7 q^{50} - 5 q^{52} + 10 q^{53} - 2 q^{55} + 24 q^{56} - 16 q^{59} + 8 q^{61} - 2 q^{62} - 15 q^{64} - 2 q^{65} + 28 q^{67} + 2 q^{70} + 2 q^{71} + 8 q^{73} + 36 q^{74} - 2 q^{76} - 2 q^{77} + 20 q^{79} + 18 q^{80} - 46 q^{82} - 24 q^{83} - 6 q^{85} + 12 q^{86} - 3 q^{88} + 20 q^{89} - 2 q^{91} + 8 q^{92} - 14 q^{94} + 22 q^{95} - 12 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.08613
0.571993
2.51414
−2.08613 0 2.35194 −2.43807 0 1.35194 −0.734191 0 5.08613
1.2 0.571993 0 −1.67282 4.24482 0 −2.67282 −2.10083 0 2.42801
1.3 2.51414 0 4.32088 0.193252 0 3.32088 5.83502 0 0.485863
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1287.2.a.j 3
3.b odd 2 1 429.2.a.e 3
12.b even 2 1 6864.2.a.bu 3
33.d even 2 1 4719.2.a.u 3
39.d odd 2 1 5577.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.e 3 3.b odd 2 1
1287.2.a.j 3 1.a even 1 1 trivial
4719.2.a.u 3 33.d even 2 1
5577.2.a.l 3 39.d odd 2 1
6864.2.a.bu 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1287))\):

\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 10T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 5T + 3 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 10 T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$19$ \( T^{3} - 10 T^{2} + 24 T - 12 \) Copy content Toggle raw display
$23$ \( T^{3} - 12 T^{2} + 24 T + 68 \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + 12 T - 162 \) Copy content Toggle raw display
$31$ \( T^{3} - 16 T^{2} + 74 T - 86 \) Copy content Toggle raw display
$37$ \( T^{3} - 96T + 288 \) Copy content Toggle raw display
$41$ \( T^{3} - 108T + 244 \) Copy content Toggle raw display
$43$ \( T^{3} + 16 T^{2} + 36 T - 162 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} - 44 T + 72 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} + 12 T + 24 \) Copy content Toggle raw display
$59$ \( T^{3} + 16 T^{2} + 40 T - 48 \) Copy content Toggle raw display
$61$ \( T^{3} - 8T^{2} + 48 \) Copy content Toggle raw display
$67$ \( T^{3} - 28 T^{2} + 214 T - 246 \) Copy content Toggle raw display
$71$ \( T^{3} - 2 T^{2} - 36 T - 24 \) Copy content Toggle raw display
$73$ \( T^{3} - 8 T^{2} - 188 T + 1692 \) Copy content Toggle raw display
$79$ \( T^{3} - 20 T^{2} + 128 T - 262 \) Copy content Toggle raw display
$83$ \( T^{3} + 24 T^{2} + 120 T + 144 \) Copy content Toggle raw display
$89$ \( T^{3} - 20 T^{2} + 54 T + 498 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} - 48 T - 608 \) Copy content Toggle raw display
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