Properties

Label 4012.2.a.f
Level $4012$
Weight $2$
Character orbit 4012.a
Self dual yes
Analytic conductor $32.036$
Analytic rank $1$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_1 q^{5} + q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_1 q^{5} + q^{7} - 2 q^{9} - 2 q^{11} + (\beta_1 + 1) q^{13} + \beta_1 q^{15} + q^{17} + (\beta_1 - 2) q^{19} - q^{21} + ( - \beta_{2} - \beta_1) q^{23} + (2 \beta_{2} + 6) q^{25} + 5 q^{27} + ( - \beta_1 - 2) q^{29} + ( - \beta_{2} + \beta_1 + 2) q^{31} + 2 q^{33} - \beta_1 q^{35} + ( - \beta_{2} + 2 \beta_1 + 3) q^{37} + ( - \beta_1 - 1) q^{39} + (\beta_1 - 2) q^{41} + ( - \beta_{2} + \beta_1 - 2) q^{43} + 2 \beta_1 q^{45} + (\beta_{2} + \beta_1 - 4) q^{47} - 6 q^{49} - q^{51} + (2 \beta_{2} + 5) q^{53} + 2 \beta_1 q^{55} + ( - \beta_1 + 2) q^{57} - q^{59} + ( - \beta_1 - 3) q^{61} - 2 q^{63} + ( - 2 \beta_{2} - \beta_1 - 11) q^{65} + (2 \beta_{2} + 2 \beta_1 - 2) q^{67} + (\beta_{2} + \beta_1) q^{69} + (2 \beta_{2} - 2) q^{71} + 2 q^{73} + ( - 2 \beta_{2} - 6) q^{75} - 2 q^{77} - 5 q^{79} + q^{81} + ( - \beta_{2} + 2 \beta_1 - 1) q^{83} - \beta_1 q^{85} + (\beta_1 + 2) q^{87} + ( - \beta_{2} + \beta_1 + 4) q^{89} + (\beta_1 + 1) q^{91} + (\beta_{2} - \beta_1 - 2) q^{93} + ( - 2 \beta_{2} + 2 \beta_1 - 11) q^{95} + ( - 2 \beta_{2} + \beta_1 - 3) q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 3 q^{21} + 20 q^{25} + 15 q^{27} - 5 q^{29} + 4 q^{31} + 6 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 7 q^{41} - 8 q^{43} - 2 q^{45} - 12 q^{47} - 18 q^{49} - 3 q^{51} + 17 q^{53} - 2 q^{55} + 7 q^{57} - 3 q^{59} - 8 q^{61} - 6 q^{63} - 34 q^{65} - 6 q^{67} - 4 q^{71} + 6 q^{73} - 20 q^{75} - 6 q^{77} - 15 q^{79} + 3 q^{81} - 6 q^{83} + q^{85} + 5 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} - 37 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2\nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
2.46050
0.239123
−1.69963
0 −1.00000 0 −3.92101 0 1.00000 0 −2.00000 0
1.2 0 −1.00000 0 0.521753 0 1.00000 0 −2.00000 0
1.3 0 −1.00000 0 4.39926 0 1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

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 4012.2.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.f 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - T_{5}^{2} - 17T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} - 17T + 9 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( (T + 2)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 16 T + 8 \) Copy content Toggle raw display
$17$ \( (T - 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 7T^{2} - T - 31 \) Copy content Toggle raw display
$23$ \( T^{3} - 36T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} - 9 T - 21 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 44 T + 168 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 96 T + 632 \) Copy content Toggle raw display
$41$ \( T^{3} + 7T^{2} - T - 31 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 28 T - 8 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} + 12 T - 88 \) Copy content Toggle raw display
$53$ \( T^{3} - 17 T^{2} - 5 T + 717 \) Copy content Toggle raw display
$59$ \( (T + 1)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} + 8 T^{2} + 4 T - 24 \) Copy content Toggle raw display
$67$ \( T^{3} + 6 T^{2} - 132 T - 344 \) Copy content Toggle raw display
$71$ \( T^{3} + 4 T^{2} - 96 T + 192 \) Copy content Toggle raw display
$73$ \( (T - 2)^{3} \) Copy content Toggle raw display
$79$ \( (T + 5)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} + 6 T^{2} - 96 T + 216 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} - 16 T + 232 \) Copy content Toggle raw display
$97$ \( T^{3} + 12 T^{2} - 84 T - 392 \) Copy content Toggle raw display
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