Properties

Label 4018.2.a.bl
Level $4018$
Weight $2$
Character orbit 4018.a
Self dual yes
Analytic conductor $32.084$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.37108.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + 5x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 7x^{2} + 5x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) 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
2.48604
−2.31364
1.66364
−0.836038
1.00000 −3.18039 1.00000 1.48604 −3.18039 0 1.00000 7.11489 1.48604
1.2 1.00000 −2.35295 1.00000 −3.31364 −2.35295 0 1.00000 2.53635 −3.31364
1.3 1.00000 0.232297 1.00000 0.663642 0.232297 0 1.00000 −2.94604 0.663642
1.4 1.00000 2.30104 1.00000 −1.83604 2.30104 0 1.00000 2.29479 −1.83604
\(n\): 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.bl 4
7.b odd 2 1 4018.2.a.bm yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.bl 4 1.a even 1 1 trivial
4018.2.a.bm yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{4} + 3T_{3}^{3} - 6T_{3}^{2} - 16T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 3T_{5}^{3} - 4T_{5}^{2} - 8T_{5} + 6 \) Copy content Toggle raw display
\( T_{11}^{4} - 20T_{11}^{2} - 10T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} - 6 T^{2} - 16 T + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} - 4 T^{2} - 8 T + 6 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 20 T^{2} - 10 T + 8 \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 20 T^{2} - 36 T - 56 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} - 32 T^{2} - 64 T - 28 \) Copy content Toggle raw display
$19$ \( T^{4} + 14 T^{3} + 56 T^{2} + 60 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} - 40 T^{2} - 68 T + 32 \) Copy content Toggle raw display
$29$ \( T^{4} + 3 T^{3} - 22 T^{2} - 2 T + 6 \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} - 22 T^{2} - 28 T - 8 \) Copy content Toggle raw display
$37$ \( T^{4} + 14 T^{3} + 36 T^{2} + \cdots - 224 \) Copy content Toggle raw display
$41$ \( (T + 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} - 60 T^{2} + 256 T - 288 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 92 T^{2} - 380 T - 448 \) Copy content Toggle raw display
$53$ \( T^{4} - 11 T^{3} - 100 T^{2} + \cdots + 446 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} - 82 T^{2} - 206 T - 84 \) Copy content Toggle raw display
$61$ \( T^{4} + 15 T^{3} + 18 T^{2} + \cdots - 162 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} - 76 T^{2} + \cdots + 1204 \) Copy content Toggle raw display
$71$ \( T^{4} - 15 T^{3} + 56 T^{2} - 56 T + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} - 64 T^{2} - 224 T + 512 \) Copy content Toggle raw display
$79$ \( T^{4} - 15 T^{3} - 24 T^{2} + \cdots + 768 \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} - 46 T^{2} + \cdots - 2192 \) Copy content Toggle raw display
$89$ \( T^{4} + 35 T^{3} + 434 T^{2} + \cdots + 3992 \) Copy content Toggle raw display
$97$ \( T^{4} - 5 T^{3} - 186 T^{2} + \cdots + 3832 \) Copy content Toggle raw display
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