Properties

Label 2001.2.a.g
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - 2 q^{4} - \beta_1 q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 2 q^{4} - \beta_1 q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7} + q^{9} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + 2 q^{12} + (2 \beta_{2} + 1) q^{13} + \beta_1 q^{15} + 4 q^{16} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{17} + (\beta_{3} + 2 \beta_{2} + 2) q^{19} + 2 \beta_1 q^{20} + ( - \beta_{3} + \beta_{2} + 1) q^{21} + q^{23} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{25} - q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{28} + q^{29} + (\beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{31} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{33} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{35} - 2 q^{36} + ( - \beta_{2} + 3 \beta_1 - 1) q^{37} + ( - 2 \beta_{2} - 1) q^{39} + ( - 2 \beta_{3} - \beta_1 + 2) q^{41} + ( - \beta_{2} - 3 \beta_1 + 1) q^{43} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{44} - \beta_1 q^{45} + \beta_1 q^{47} - 4 q^{48} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{49} + (2 \beta_{3} - \beta_{2} - 1) q^{51} + ( - 4 \beta_{2} - 2) q^{52} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{53} + (2 \beta_{3} - 2) q^{55} + ( - \beta_{3} - 2 \beta_{2} - 2) q^{57} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{59} - 2 \beta_1 q^{60} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{61} + (\beta_{3} - \beta_{2} - 1) q^{63} - 8 q^{64} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{65} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{67} + (4 \beta_{3} - 2 \beta_{2} - 2) q^{68} - q^{69} + (2 \beta_{3} - \beta_1 - 5) q^{71} + ( - \beta_{3} + \beta_{2} - 5) q^{73} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{75} + ( - 2 \beta_{3} - 4 \beta_{2} - 4) q^{76} + (2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{77} + (3 \beta_{3} + 2 \beta_1 - 6) q^{79} - 4 \beta_1 q^{80} + q^{81} + ( - 4 \beta_{2} + 3 \beta_1 - 6) q^{83} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{84} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{85} - q^{87} + (3 \beta_{2} - 6 \beta_1 + 1) q^{89} + ( - \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 11) q^{91} - 2 q^{92} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{93} + ( - 4 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{95} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{97} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9} + 8 q^{12} + 2 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{21} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{28} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 8 q^{36} + 4 q^{37} + 6 q^{41} - 2 q^{45} + 2 q^{47} - 16 q^{48} + 4 q^{49} - 2 q^{51} - 8 q^{53} - 8 q^{55} - 4 q^{57} - 22 q^{59} - 4 q^{60} - 16 q^{61} - 2 q^{63} - 32 q^{64} - 10 q^{65} + 4 q^{67} - 4 q^{68} - 4 q^{69} - 22 q^{71} - 22 q^{73} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 20 q^{79} - 8 q^{80} + 4 q^{81} - 10 q^{83} - 4 q^{84} - 2 q^{85} - 4 q^{87} - 14 q^{89} - 26 q^{91} - 8 q^{92} - 2 q^{93} - 14 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 2\beta_{2} + 3\beta _1 - 1 \) 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
2.37988
−0.751024
−1.92022
0.291367
0 −1.00000 −2.00000 −2.95969 0 1.80007 0 1.00000 0
1.2 0 −1.00000 −2.00000 −1.58049 0 −3.08254 0 1.00000 0
1.3 0 −1.00000 −2.00000 0.399447 0 −3.44099 0 1.00000 0
1.4 0 −1.00000 −2.00000 2.14073 0 2.72347 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 2001.2.a.g 4
3.b odd 2 1 6003.2.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.g 4 1.a even 1 1 trivial
6003.2.a.g 4 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - 6 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 14 T^{2} - 16 T + 52 \) Copy content Toggle raw display
$11$ \( T^{4} - 28 T^{2} - 48 T - 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 38 T^{2} - 24 T + 125 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 44 T^{2} + 22 T + 355 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} - 56 T^{2} + 116 T + 835 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} - 38 T^{2} - 48 T + 4 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} - 56 T^{2} + 108 T + 835 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} - 46 T^{2} + 288 T - 364 \) Copy content Toggle raw display
$43$ \( T^{4} - 92 T^{2} - 216 T - 73 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 6 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} - 56 T^{2} - 416 T - 304 \) Copy content Toggle raw display
$59$ \( T^{4} + 22 T^{3} + 130 T^{2} + \cdots + 133 \) Copy content Toggle raw display
$61$ \( T^{4} + 16 T^{3} + 8 T^{2} + \cdots - 2240 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} - 56 T^{2} + 128 T + 832 \) Copy content Toggle raw display
$71$ \( T^{4} + 22 T^{3} + 130 T^{2} + \cdots + 197 \) Copy content Toggle raw display
$73$ \( T^{4} + 22 T^{3} + 166 T^{2} + \cdots + 508 \) Copy content Toggle raw display
$79$ \( T^{4} + 20 T^{3} - 1676 T - 7133 \) Copy content Toggle raw display
$83$ \( T^{4} + 10 T^{3} - 122 T^{2} + \cdots - 68 \) Copy content Toggle raw display
$89$ \( T^{4} + 14 T^{3} - 192 T^{2} + \cdots + 10207 \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} - 24 T^{2} - 32 T + 64 \) Copy content Toggle raw display
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