Properties

Label 8673.2.a.s
Level $8673$
Weight $2$
Character orbit 8673.a
Self dual yes
Analytic conductor $69.254$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8673,2,Mod(1,8673)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8673, 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("8673.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8673 = 3 \cdot 7^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8673.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: \(69.2542536731\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1 + 1) q^{5} + \beta_1 q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1 + 1) q^{5} + \beta_1 q^{6} + q^{8} + q^{9} + ( - \beta_{2} + 2 \beta_1 - 2) q^{10} + ( - \beta_{2} - \beta_1 - 1) q^{11} + (\beta_{2} + 1) q^{12} + (\beta_{2} + \beta_1 - 1) q^{13} + (\beta_{2} - \beta_1 + 1) q^{15} + ( - 2 \beta_{2} + \beta_1 - 2) q^{16} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{17} + \beta_1 q^{18} + (\beta_{2} - 2) q^{19} + ( - \beta_1 + 3) q^{20} + ( - \beta_{2} - 2 \beta_1 - 4) q^{22} + ( - \beta_{2} - 2 \beta_1) q^{23} + q^{24} + (\beta_{2} - 3 \beta_1) q^{25} + (\beta_{2} + 4) q^{26} + q^{27} + (2 \beta_{2} + \beta_1 - 3) q^{29} + ( - \beta_{2} + 2 \beta_1 - 2) q^{30} + ( - 2 \beta_{2} + \beta_1 - 5) q^{31} + (\beta_{2} - 4 \beta_1 - 1) q^{32} + ( - \beta_{2} - \beta_1 - 1) q^{33} + (2 \beta_{2} - 5 \beta_1 + 3) q^{34} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} - 2 \beta_1 - 2) q^{37} + ( - \beta_1 + 1) q^{38} + (\beta_{2} + \beta_1 - 1) q^{39} + (\beta_{2} - \beta_1 + 1) q^{40} + (2 \beta_{2} - 3 \beta_1 + 1) q^{41} + ( - 3 \beta_{2} + 5 \beta_1 + 1) q^{43} + ( - 3 \beta_1 - 5) q^{44} + (\beta_{2} - \beta_1 + 1) q^{45} + ( - 2 \beta_{2} - \beta_1 - 7) q^{46} + ( - 4 \beta_{2} + \beta_1 - 5) q^{47} + ( - 2 \beta_{2} + \beta_1 - 2) q^{48} + ( - 3 \beta_{2} + \beta_1 - 8) q^{50} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{51} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{52} + (\beta_{2} - 5 \beta_1 + 1) q^{53} + \beta_1 q^{54} + (\beta_{2} - \beta_1 - 1) q^{55} + (\beta_{2} - 2) q^{57} + (\beta_{2} - \beta_1 + 5) q^{58} - q^{59} + ( - \beta_1 + 3) q^{60} + ( - 4 \beta_{2} - \beta_1 - 1) q^{61} + (\beta_{2} - 7 \beta_1 + 1) q^{62} + ( - 2 \beta_1 - 7) q^{64} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{65} + ( - \beta_{2} - 2 \beta_1 - 4) q^{66} + (5 \beta_{2} + \beta_1 + 5) q^{67} + (\beta_{2} + \beta_1 - 9) q^{68} + ( - \beta_{2} - 2 \beta_1) q^{69} + (\beta_{2} - 3 \beta_1 + 9) q^{71} + q^{72} + (4 \beta_{2} + 3 \beta_1 - 1) q^{73} + ( - 2 \beta_{2} - 3 \beta_1 - 7) q^{74} + (\beta_{2} - 3 \beta_1) q^{75} + ( - 3 \beta_{2} + \beta_1 + 1) q^{76} + (\beta_{2} + 4) q^{78} + (\beta_{2} - 3 \beta_1 + 1) q^{79} + ( - \beta_{2} + 4 \beta_1 - 8) q^{80} + q^{81} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{82} + ( - 6 \beta_{2} + 5 \beta_1 - 1) q^{83} + ( - \beta_{2} + 6 \beta_1 - 12) q^{85} + (5 \beta_{2} - 2 \beta_1 + 12) q^{86} + (2 \beta_{2} + \beta_1 - 3) q^{87} + ( - \beta_{2} - \beta_1 - 1) q^{88} + (4 \beta_{2} - 3 \beta_1 + 9) q^{89} + ( - \beta_{2} + 2 \beta_1 - 2) q^{90} + (\beta_{2} - 5 \beta_1 - 5) q^{92} + ( - 2 \beta_{2} + \beta_1 - 5) q^{93} + (\beta_{2} - 9 \beta_1 - 1) q^{94} + ( - 3 \beta_{2} + 2 \beta_1) q^{95} + (\beta_{2} - 4 \beta_1 - 1) q^{96} + ( - \beta_{2} + 5 \beta_1 - 5) q^{97} + ( - \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 2 q^{4} + 2 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 2 q^{4} + 2 q^{5} + 3 q^{8} + 3 q^{9} - 5 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} + 2 q^{15} - 4 q^{16} - 3 q^{17} - 7 q^{19} + 9 q^{20} - 11 q^{22} + q^{23} + 3 q^{24} - q^{25} + 11 q^{26} + 3 q^{27} - 11 q^{29} - 5 q^{30} - 13 q^{31} - 4 q^{32} - 2 q^{33} + 7 q^{34} + 2 q^{36} - 5 q^{37} + 3 q^{38} - 4 q^{39} + 2 q^{40} + q^{41} + 6 q^{43} - 15 q^{44} + 2 q^{45} - 19 q^{46} - 11 q^{47} - 4 q^{48} - 21 q^{50} - 3 q^{51} + 11 q^{52} + 2 q^{53} - 4 q^{55} - 7 q^{57} + 14 q^{58} - 3 q^{59} + 9 q^{60} + q^{61} + 2 q^{62} - 21 q^{64} - 11 q^{66} + 10 q^{67} - 28 q^{68} + q^{69} + 26 q^{71} + 3 q^{72} - 7 q^{73} - 19 q^{74} - q^{75} + 6 q^{76} + 11 q^{78} + 2 q^{79} - 23 q^{80} + 3 q^{81} - 18 q^{82} + 3 q^{83} - 35 q^{85} + 31 q^{86} - 11 q^{87} - 2 q^{88} + 23 q^{89} - 5 q^{90} - 16 q^{92} - 13 q^{93} - 4 q^{94} + 3 q^{95} - 4 q^{96} - 14 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
−1.86081
−0.254102
2.11491
−1.86081 1.00000 1.46260 3.32340 −1.86081 0 1.00000 1.00000 −6.18421
1.2 −0.254102 1.00000 −1.93543 −1.68133 −0.254102 0 1.00000 1.00000 0.427229
1.3 2.11491 1.00000 2.47283 0.357926 2.11491 0 1.00000 1.00000 0.756981
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(59\) \(1\)

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 8673.2.a.s 3
7.b odd 2 1 177.2.a.d 3
21.c even 2 1 531.2.a.d 3
28.d even 2 1 2832.2.a.t 3
35.c odd 2 1 4425.2.a.w 3
84.h odd 2 1 8496.2.a.bl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.d 3 7.b odd 2 1
531.2.a.d 3 21.c even 2 1
2832.2.a.t 3 28.d even 2 1
4425.2.a.w 3 35.c odd 2 1
8496.2.a.bl 3 84.h odd 2 1
8673.2.a.s 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8673))\):

\( T_{2}^{3} - 4T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 5T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 11T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 4T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 5 T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 11 T + 4 \) Copy content Toggle raw display
$13$ \( T^{3} + 4 T^{2} - 7 T - 26 \) Copy content Toggle raw display
$17$ \( T^{3} + 3 T^{2} - 43 T - 98 \) Copy content Toggle raw display
$19$ \( T^{3} + 7 T^{2} + 11 T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} - 27 T + 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 11 T^{2} + 9 T - 74 \) Copy content Toggle raw display
$31$ \( T^{3} + 13 T^{2} + 37 T - 28 \) Copy content Toggle raw display
$37$ \( T^{3} + 5 T^{2} - 19 T + 14 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 39 T - 74 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 91 T + 592 \) Copy content Toggle raw display
$47$ \( T^{3} + 11 T^{2} - 37 T - 496 \) Copy content Toggle raw display
$53$ \( T^{3} - 2 T^{2} - 89 T - 58 \) Copy content Toggle raw display
$59$ \( (T + 1)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} - T^{2} - 101 T - 98 \) Copy content Toggle raw display
$67$ \( T^{3} - 10 T^{2} - 119 T + 784 \) Copy content Toggle raw display
$71$ \( T^{3} - 26 T^{2} + 193 T - 424 \) Copy content Toggle raw display
$73$ \( T^{3} + 7 T^{2} - 141 T - 718 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 31 T - 32 \) Copy content Toggle raw display
$83$ \( T^{3} - 3 T^{2} - 199 T + 148 \) Copy content Toggle raw display
$89$ \( T^{3} - 23 T^{2} + 91 T + 278 \) Copy content Toggle raw display
$97$ \( T^{3} + 14 T^{2} - 25 T - 202 \) Copy content Toggle raw display
show more
show less