Properties

Label 1984.2.a.u
Level $1984$
Weight $2$
Character orbit 1984.a
Self dual yes
Analytic conductor $15.842$
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] = [1984,2,Mod(1,1984)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1984, 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("1984.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1984 = 2^{6} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1984.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: \(15.8423197610\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 992)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} - \beta_{2} q^{5} + ( - \beta_1 + 3) q^{7} + ( - 2 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} - \beta_{2} q^{5} + ( - \beta_1 + 3) q^{7} + ( - 2 \beta_1 + 3) q^{9} + ( - 2 \beta_1 + 2) q^{11} + ( - \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{2} + \beta_1 + 3) q^{15} + (\beta_{2} - \beta_1 - 1) q^{17} + ( - 3 \beta_1 + 1) q^{19} + ( - 3 \beta_{2} + 5 \beta_1 - 5) q^{21} + (\beta_{2} - \beta_1 + 5) q^{23} + ( - \beta_{2} + 2 \beta_1 - 1) q^{25} + (4 \beta_1 - 4) q^{27} + (\beta_{2} - 3 \beta_1 + 1) q^{29} - q^{31} + ( - 2 \beta_{2} + 6 \beta_1 - 6) q^{33} + ( - 2 \beta_{2} + \beta_1 + 1) q^{35} - 2 q^{37} + ( - 2 \beta_{2} + 4) q^{39} + ( - \beta_{2} + 2 \beta_1 - 6) q^{41} + (3 \beta_{2} - \beta_1 - 3) q^{43} + ( - \beta_{2} + 2 \beta_1 + 2) q^{45} + ( - 2 \beta_{2} + 6 \beta_1 + 2) q^{47} + (\beta_{2} - 6 \beta_1 + 5) q^{49} + (2 \beta_{2} - 4) q^{51} + (2 \beta_{2} + 2) q^{53} + (2 \beta_1 + 2) q^{55} + ( - \beta_{2} + 7 \beta_1 - 7) q^{57} + (5 \beta_1 + 1) q^{59} + (3 \beta_{2} - 3 \beta_1 - 3) q^{61} + (2 \beta_{2} - 9 \beta_1 + 15) q^{63} + ( - 3 \beta_{2} + \beta_1 + 3) q^{65} + (2 \beta_1 - 6) q^{67} + ( - 4 \beta_{2} + 6 \beta_1 - 10) q^{69} + (4 \beta_{2} - \beta_1 + 3) q^{71} + ( - 4 \beta_{2} + 2) q^{73} + ( - 4 \beta_1 + 8) q^{75} + (2 \beta_{2} - 8 \beta_1 + 12) q^{77} + ( - \beta_{2} - \beta_1 + 9) q^{79} + (4 \beta_{2} - 6 \beta_1 + 3) q^{81} + ( - 2 \beta_1 + 2) q^{83} + (3 \beta_{2} - \beta_1 - 3) q^{85} + (6 \beta_1 - 10) q^{87} + ( - 3 \beta_{2} + \beta_1 + 1) q^{89} + ( - 3 \beta_{2} + 3 \beta_1 + 1) q^{91} + (\beta_{2} - \beta_1 + 1) q^{93} + (2 \beta_{2} + 3 \beta_1 + 3) q^{95} + (3 \beta_{2} + 2 \beta_1 - 6) q^{97} + (4 \beta_{2} - 10 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 8 q^{7} + 7 q^{9} + 4 q^{11} + 4 q^{13} + 10 q^{15} - 4 q^{17} - 10 q^{21} + 14 q^{23} - q^{25} - 8 q^{27} - 3 q^{31} - 12 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} - 16 q^{41} - 10 q^{43} + 8 q^{45} + 12 q^{47} + 9 q^{49} - 12 q^{51} + 6 q^{53} + 8 q^{55} - 14 q^{57} + 8 q^{59} - 12 q^{61} + 36 q^{63} + 10 q^{65} - 16 q^{67} - 24 q^{69} + 8 q^{71} + 6 q^{73} + 20 q^{75} + 28 q^{77} + 26 q^{79} + 3 q^{81} + 4 q^{83} - 10 q^{85} - 24 q^{87} + 4 q^{89} + 6 q^{91} + 2 q^{93} + 12 q^{95} - 16 q^{97} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : 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
−1.81361
2.34292
0.470683
0 −3.10278 0 −0.289169 0 4.81361 0 6.62721 0
1.2 0 −1.14637 0 −2.48929 0 0.657077 0 −1.68585 0
1.3 0 2.24914 0 2.77846 0 2.52932 0 2.05863 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(31\) \(1\)

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 1984.2.a.u 3
4.b odd 2 1 1984.2.a.w 3
8.b even 2 1 992.2.a.d yes 3
8.d odd 2 1 992.2.a.c 3
24.f even 2 1 8928.2.a.bc 3
24.h odd 2 1 8928.2.a.bd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
992.2.a.c 3 8.d odd 2 1
992.2.a.d yes 3 8.b even 2 1
1984.2.a.u 3 1.a even 1 1 trivial
1984.2.a.w 3 4.b odd 2 1
8928.2.a.bc 3 24.f even 2 1
8928.2.a.bd 3 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1984))\):

\( T_{3}^{3} + 2T_{3}^{2} - 6T_{3} - 8 \) Copy content Toggle raw display
\( T_{5}^{3} - 7T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 8T_{7}^{2} + 17T_{7} - 8 \) Copy content Toggle raw display
\( T_{19}^{3} - 39T_{19} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 6 T - 8 \) Copy content Toggle raw display
$5$ \( T^{3} - 7T - 2 \) Copy content Toggle raw display
$7$ \( T^{3} - 8 T^{2} + 17 T - 8 \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} - 12 T + 16 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 2 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 2 T - 4 \) Copy content Toggle raw display
$19$ \( T^{3} - 39T - 16 \) Copy content Toggle raw display
$23$ \( T^{3} - 14 T^{2} + 58 T - 64 \) Copy content Toggle raw display
$29$ \( T^{3} - 34T - 76 \) Copy content Toggle raw display
$31$ \( (T + 1)^{3} \) Copy content Toggle raw display
$37$ \( (T + 2)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 16 T^{2} + 69 T + 86 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} - 22 T - 8 \) Copy content Toggle raw display
$47$ \( T^{3} - 12 T^{2} - 88 T + 1088 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 16 T + 64 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 87 T + 344 \) Copy content Toggle raw display
$61$ \( T^{3} + 12 T^{2} - 18 T - 108 \) Copy content Toggle raw display
$67$ \( T^{3} + 16 T^{2} + 68 T + 64 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 79 T + 544 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 100 T + 88 \) Copy content Toggle raw display
$79$ \( T^{3} - 26 T^{2} + 210 T - 496 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 12 T + 16 \) Copy content Toggle raw display
$89$ \( T^{3} - 4 T^{2} - 50 T - 68 \) Copy content Toggle raw display
$97$ \( T^{3} + 16 T^{2} - 19 T - 722 \) Copy content Toggle raw display
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