Properties

Label 4018.2.a.bf
Level $4018$
Weight $2$
Character orbit 4018.a
Self dual yes
Analytic conductor $32.084$
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] = [4018,2,Mod(1,4018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.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: \(32.0838915322\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 574)
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.

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
−1.69963
2.46050
0.239123
1.00000 −1.00000 1.00000 −0.588364 −1.00000 0 1.00000 −2.00000 −0.588364
1.2 1.00000 −1.00000 1.00000 0.406421 −1.00000 0 1.00000 −2.00000 0.406421
1.3 1.00000 −1.00000 1.00000 4.18194 −1.00000 0 1.00000 −2.00000 4.18194
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(-1\)

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 4018.2.a.bf 3
7.b odd 2 1 4018.2.a.bh 3
7.c even 3 2 574.2.e.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.e 6 7.c even 3 2
4018.2.a.bf 3 1.a even 1 1 trivial
4018.2.a.bh 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 4T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 5 T^{2} + 4 T - 3 \) Copy content Toggle raw display
$13$ \( T^{3} - T^{2} - 25 T + 49 \) Copy content Toggle raw display
$17$ \( T^{3} + 5 T^{2} - 12 T - 63 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 24 T - 27 \) Copy content Toggle raw display
$23$ \( T^{3} + 9 T^{2} - 12 T - 79 \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + 21 T - 97 \) Copy content Toggle raw display
$31$ \( T^{3} + 11 T^{2} + 23 T - 11 \) Copy content Toggle raw display
$37$ \( T^{3} + 18 T^{2} + 99 T + 161 \) Copy content Toggle raw display
$41$ \( (T - 1)^{3} \) Copy content Toggle raw display
$43$ \( T^{3} + 13 T^{2} - 25 T - 541 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} - 107 T + 21 \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 35 T + 27 \) Copy content Toggle raw display
$59$ \( T^{3} - 7 T^{2} - 34 T + 217 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} - 119 T + 951 \) Copy content Toggle raw display
$67$ \( T^{3} + 23 T^{2} + 164 T + 343 \) Copy content Toggle raw display
$71$ \( T^{3} - 33T + 9 \) Copy content Toggle raw display
$73$ \( T^{3} + 8 T^{2} - 43 T - 257 \) Copy content Toggle raw display
$79$ \( T^{3} - T^{2} - 192 T - 9 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 45 T + 177 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 51 T - 123 \) Copy content Toggle raw display
$97$ \( T^{3} + 3 T^{2} - 78 T - 53 \) Copy content Toggle raw display
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