Properties

Label 4018.2.a.bp
Level $4018$
Weight $2$
Character orbit 4018.a
Self dual yes
Analytic conductor $32.084$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.5163008.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 5x^{2} - 6x + 1 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{3} + q^{4} + ( - \beta_{4} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{6} + q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{3} + q^{4} + ( - \beta_{4} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{6} + q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{9} + ( - \beta_{4} - 2) q^{10} + ( - \beta_{2} + 2 \beta_1 - 1) q^{11} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{12} + (\beta_{4} - \beta_{3} + \beta_{2} - 1) q^{13} + (4 \beta_{3} + 3 \beta_{2} + \beta_1) q^{15} + q^{16} + (2 \beta_{5} - \beta_{4} + \beta_1 - 3) q^{17} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{18} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{19} + ( - \beta_{4} - 2) q^{20} + ( - \beta_{2} + 2 \beta_1 - 1) q^{22} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 + 2) q^{23} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{24} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 + 3) q^{25} + (\beta_{4} - \beta_{3} + \beta_{2} - 1) q^{26} + (4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 6) q^{27} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{29} + (4 \beta_{3} + 3 \beta_{2} + \beta_1) q^{30} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{31} + q^{32} + ( - 4 \beta_{5} + 7 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 4) q^{33} + (2 \beta_{5} - \beta_{4} + \beta_1 - 3) q^{34} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{36} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 4) q^{37} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{38} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{39} + ( - \beta_{4} - 2) q^{40} + q^{41} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} + 1) q^{43} + ( - \beta_{2} + 2 \beta_1 - 1) q^{44} + (\beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 7) q^{45} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 + 2) q^{46} + (\beta_{4} + 4 \beta_{3} - \beta_1 + 3) q^{47} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{48} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 + 3) q^{50} + ( - 3 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 1) q^{51} + (\beta_{4} - \beta_{3} + \beta_{2} - 1) q^{52} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 3 \beta_1 - 3) q^{53} + (4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 6) q^{54} + (\beta_{4} + \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 5) q^{55} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 5) q^{57} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{58} + ( - 6 \beta_{5} - \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 2) q^{59} + (4 \beta_{3} + 3 \beta_{2} + \beta_1) q^{60} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 5) q^{61} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{62} + q^{64} + (2 \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{65} + ( - 4 \beta_{5} + 7 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 4) q^{66} + ( - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{2} + 3 \beta_1 - 3) q^{67} + (2 \beta_{5} - \beta_{4} + \beta_1 - 3) q^{68} + (4 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 - 10) q^{69} + ( - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 3) q^{71} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{72} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 3) q^{73} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 4) q^{74} + (4 \beta_{5} - \beta_{4} - 10 \beta_{3} - 8 \beta_{2} - 3 \beta_1 - 1) q^{75} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{76} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{78} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{79} + ( - \beta_{4} - 2) q^{80} + ( - 5 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} + 10) q^{81} + q^{82} + (2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 4 \beta_1 - 4) q^{83} + ( - 3 \beta_{5} + 5 \beta_{4} - \beta_{2} - 4 \beta_1 + 9) q^{85} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} + 1) q^{86} + ( - 2 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} + 2 \beta_1 - 4) q^{87} + ( - \beta_{2} + 2 \beta_1 - 1) q^{88} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{89} + (\beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 7) q^{90} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 + 2) q^{92} + (3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 5) q^{93} + (\beta_{4} + 4 \beta_{3} - \beta_1 + 3) q^{94} + (2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 5 \beta_1 + 2) q^{95} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{96} + (4 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 8 \beta_1) q^{97} + (3 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 7 \beta_{2} + 3 \beta_1 - 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 4 q^{3} + 6 q^{4} - 12 q^{5} - 4 q^{6} + 6 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} - 4 q^{3} + 6 q^{4} - 12 q^{5} - 4 q^{6} + 6 q^{8} + 18 q^{9} - 12 q^{10} - 4 q^{12} - 8 q^{13} - 4 q^{15} + 6 q^{16} - 16 q^{17} + 18 q^{18} - 12 q^{19} - 12 q^{20} + 12 q^{23} - 4 q^{24} + 14 q^{25} - 8 q^{26} - 28 q^{27} - 4 q^{30} + 4 q^{31} + 6 q^{32} + 20 q^{33} - 16 q^{34} + 18 q^{36} - 24 q^{37} - 12 q^{38} + 4 q^{39} - 12 q^{40} + 6 q^{41} + 4 q^{43} - 28 q^{45} + 12 q^{46} + 16 q^{47} - 4 q^{48} + 14 q^{50} - 16 q^{51} - 8 q^{52} - 16 q^{53} - 28 q^{54} + 12 q^{55} - 28 q^{57} - 16 q^{59} - 4 q^{60} - 32 q^{61} + 4 q^{62} + 6 q^{64} - 4 q^{65} + 20 q^{66} - 20 q^{67} - 16 q^{68} - 56 q^{69} + 12 q^{71} + 18 q^{72} - 16 q^{73} - 24 q^{74} + 4 q^{75} - 12 q^{76} + 4 q^{78} - 12 q^{80} + 42 q^{81} + 6 q^{82} - 32 q^{83} + 48 q^{85} + 4 q^{86} - 20 q^{87} - 8 q^{89} - 28 q^{90} + 12 q^{92} - 12 q^{93} + 16 q^{94} + 28 q^{95} - 4 q^{96} - 8 q^{97} - 76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 5x^{2} - 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 6\nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 8\nu^{2} + 5\nu - 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} - 3\nu^{4} - 11\nu^{3} + 10\nu^{2} + 13\nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 5\beta_{4} - 7\beta_{3} + \beta_{2} + 8\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 8\beta_{4} - 11\beta_{3} + 7\beta_{2} + 28\beta _1 + 10 \) 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
−1.14577
0.218114
0.545336
2.56754
1.60043
−1.78566
1.00000 −3.30757 1.00000 −3.87277 −3.30757 0 1.00000 7.94005 −3.87277
1.2 1.00000 −3.26089 1.00000 1.58475 −3.26089 0 1.00000 7.63338 1.58475
1.3 1.00000 −1.93139 1.00000 −1.16627 −1.93139 0 1.00000 0.730254 −1.16627
1.4 1.00000 −0.0387751 1.00000 −2.61052 −0.0387751 0 1.00000 −2.99850 −2.61052
1.5 1.00000 1.82475 1.00000 −2.37517 1.82475 0 1.00000 0.329701 −2.37517
1.6 1.00000 2.71387 1.00000 −3.56002 2.71387 0 1.00000 4.36512 −3.56002
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bp 6
7.b odd 2 1 4018.2.a.bq yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.bp 6 1.a even 1 1 trivial
4018.2.a.bq yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{6} + 4T_{3}^{5} - 10T_{3}^{4} - 44T_{3}^{3} + 20T_{3}^{2} + 104T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{6} + 12T_{5}^{5} + 50T_{5}^{4} + 68T_{5}^{3} - 68T_{5}^{2} - 248T_{5} - 158 \) Copy content Toggle raw display
\( T_{11}^{6} - 50T_{11}^{4} + 8T_{11}^{3} + 498T_{11}^{2} - 300T_{11} - 398 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 4 T^{5} - 10 T^{4} - 44 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} + 12 T^{5} + 50 T^{4} + \cdots - 158 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 50 T^{4} + 8 T^{3} + 498 T^{2} + \cdots - 398 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} - 6 T^{4} - 116 T^{3} + \cdots - 476 \) Copy content Toggle raw display
$17$ \( T^{6} + 16 T^{5} + 54 T^{4} + \cdots + 2212 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + 14 T^{4} + \cdots + 196 \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{5} - 4 T^{4} + \cdots + 1828 \) Copy content Toggle raw display
$29$ \( T^{6} - 78 T^{4} + 48 T^{3} + \cdots - 3374 \) Copy content Toggle raw display
$31$ \( T^{6} - 4 T^{5} - 90 T^{4} + 128 T^{3} + \cdots + 968 \) Copy content Toggle raw display
$37$ \( T^{6} + 24 T^{5} + 176 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T - 1)^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - 4 T^{5} - 108 T^{4} + \cdots - 18704 \) Copy content Toggle raw display
$47$ \( T^{6} - 16 T^{5} - 26 T^{4} + \cdots + 12964 \) Copy content Toggle raw display
$53$ \( T^{6} + 16 T^{5} - 84 T^{4} + \cdots - 7822 \) Copy content Toggle raw display
$59$ \( T^{6} + 16 T^{5} - 234 T^{4} + \cdots - 674318 \) Copy content Toggle raw display
$61$ \( T^{6} + 32 T^{5} + 360 T^{4} + \cdots - 29102 \) Copy content Toggle raw display
$67$ \( T^{6} + 20 T^{5} - 40 T^{4} + \cdots + 279314 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} - 92 T^{4} + \cdots + 91424 \) Copy content Toggle raw display
$73$ \( T^{6} + 16 T^{5} + 74 T^{4} + \cdots + 1016 \) Copy content Toggle raw display
$79$ \( T^{6} - 88 T^{4} - 288 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$83$ \( T^{6} + 32 T^{5} + 134 T^{4} + \cdots + 515234 \) Copy content Toggle raw display
$89$ \( T^{6} + 8 T^{5} - 128 T^{4} + \cdots - 10256 \) Copy content Toggle raw display
$97$ \( T^{6} + 8 T^{5} - 440 T^{4} + \cdots - 1106816 \) Copy content Toggle raw display
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