Properties

Label 867.4.a.k
Level $867$
Weight $4$
Character orbit 867.a
Self dual yes
Analytic conductor $51.155$
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] = [867,4,Mod(1,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 867.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: \(51.1546559750\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5912.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 14x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} - 3 q^{3} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + (\beta_{2} - 2 \beta_1 - 2) q^{5} + (3 \beta_1 - 6) q^{6} + (4 \beta_{2} - 4 \beta_1 + 4) q^{7} + (5 \beta_{2} + 11) q^{8} + 9 q^{9}+ \cdots + ( - 27 \beta_{2} - 90 \beta_1 - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{2} - 9 q^{3} + 13 q^{4} - 8 q^{5} - 15 q^{6} + 8 q^{7} + 33 q^{8} + 27 q^{9} + 38 q^{10} - 34 q^{11} - 39 q^{12} + 36 q^{13} + 104 q^{14} + 24 q^{15} - 79 q^{16} + 45 q^{18} - 142 q^{19} + 126 q^{20}+ \cdots - 306 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 14x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 9 \) 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
4.55528
−0.795427
−2.75985
−2.55528 −3.00000 −1.47057 −8.47057 7.66583 −3.66117 24.1999 9.00000 21.6446
1.2 2.79543 −3.00000 −0.185590 −7.18559 −8.38628 −19.9241 −22.8822 9.00000 −20.0868
1.3 4.75985 −3.00000 14.6562 7.65616 −14.2795 31.5852 31.6823 9.00000 36.4422
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(17\) \( +1 \)

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 867.4.a.k 3
17.b even 2 1 51.4.a.e 3
51.c odd 2 1 153.4.a.f 3
68.d odd 2 1 816.4.a.s 3
85.c even 2 1 1275.4.a.q 3
119.d odd 2 1 2499.4.a.n 3
204.h even 2 1 2448.4.a.bd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.a.e 3 17.b even 2 1
153.4.a.f 3 51.c odd 2 1
816.4.a.s 3 68.d odd 2 1
867.4.a.k 3 1.a even 1 1 trivial
1275.4.a.q 3 85.c even 2 1
2448.4.a.bd 3 204.h even 2 1
2499.4.a.n 3 119.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(867))\):

\( T_{2}^{3} - 5T_{2}^{2} - 6T_{2} + 34 \) Copy content Toggle raw display
\( T_{5}^{3} + 8T_{5}^{2} - 59T_{5} - 466 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 5 T^{2} + \cdots + 34 \) Copy content Toggle raw display
$3$ \( (T + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 8 T^{2} + \cdots - 466 \) Copy content Toggle raw display
$7$ \( T^{3} - 8 T^{2} + \cdots - 2304 \) Copy content Toggle raw display
$11$ \( T^{3} + 34 T^{2} + \cdots + 8964 \) Copy content Toggle raw display
$13$ \( T^{3} - 36 T^{2} + \cdots + 122698 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 142 T^{2} + \cdots + 8244 \) Copy content Toggle raw display
$23$ \( T^{3} + 110 T^{2} + \cdots + 53240 \) Copy content Toggle raw display
$29$ \( T^{3} + 90 T^{2} + \cdots + 415320 \) Copy content Toggle raw display
$31$ \( T^{3} - 148 T^{2} + \cdots + 640448 \) Copy content Toggle raw display
$37$ \( T^{3} + 110 T^{2} + \cdots + 5969792 \) Copy content Toggle raw display
$41$ \( T^{3} + 720 T^{2} + \cdots + 10440042 \) Copy content Toggle raw display
$43$ \( T^{3} + 146 T^{2} + \cdots - 62624916 \) Copy content Toggle raw display
$47$ \( T^{3} - 500 T^{2} + \cdots + 30472896 \) Copy content Toggle raw display
$53$ \( T^{3} - 610 T^{2} + \cdots + 80447688 \) Copy content Toggle raw display
$59$ \( T^{3} + 216 T^{2} + \cdots + 1302384 \) Copy content Toggle raw display
$61$ \( T^{3} - 18 T^{2} + \cdots - 8127200 \) Copy content Toggle raw display
$67$ \( T^{3} + 1404 T^{2} + \cdots + 62069312 \) Copy content Toggle raw display
$71$ \( T^{3} - 960 T^{2} + \cdots + 227624576 \) Copy content Toggle raw display
$73$ \( T^{3} - 794 T^{2} + \cdots + 227482344 \) Copy content Toggle raw display
$79$ \( T^{3} - 276 T^{2} + \cdots + 220814208 \) Copy content Toggle raw display
$83$ \( T^{3} + 1552 T^{2} + \cdots + 11261392 \) Copy content Toggle raw display
$89$ \( T^{3} - 1394 T^{2} + \cdots + 278458912 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 2026068032 \) Copy content Toggle raw display
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