Properties

Label 2002.2.a.d
Level $2002$
Weight $2$
Character orbit 2002.a
Self dual yes
Analytic conductor $15.986$
Analytic rank $0$
Dimension $2$
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] = [2002,2,Mod(1,2002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2002, 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("2002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2002 = 2 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2002.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: \(15.9860504847\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta q^{3} + q^{4} + (\beta - 1) q^{5} + \beta q^{6} - q^{7} - q^{8} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta q^{3} + q^{4} + (\beta - 1) q^{5} + \beta q^{6} - q^{7} - q^{8} + (\beta - 2) q^{9} + ( - \beta + 1) q^{10} + q^{11} - \beta q^{12} - q^{13} + q^{14} - q^{15} + q^{16} + ( - 3 \beta + 4) q^{17} + ( - \beta + 2) q^{18} + (\beta + 1) q^{19} + (\beta - 1) q^{20} + \beta q^{21} - q^{22} - 2 \beta q^{23} + \beta q^{24} + ( - \beta - 3) q^{25} + q^{26} + (4 \beta - 1) q^{27} - q^{28} + (6 \beta - 6) q^{29} + q^{30} + ( - 4 \beta + 2) q^{31} - q^{32} - \beta q^{33} + (3 \beta - 4) q^{34} + ( - \beta + 1) q^{35} + (\beta - 2) q^{36} + 4 q^{37} + ( - \beta - 1) q^{38} + \beta q^{39} + ( - \beta + 1) q^{40} + ( - 8 \beta + 4) q^{41} - \beta q^{42} + (5 \beta - 4) q^{43} + q^{44} + ( - 2 \beta + 3) q^{45} + 2 \beta q^{46} + (2 \beta + 4) q^{47} - \beta q^{48} + q^{49} + (\beta + 3) q^{50} + ( - \beta + 3) q^{51} - q^{52} + (\beta + 8) q^{53} + ( - 4 \beta + 1) q^{54} + (\beta - 1) q^{55} + q^{56} + ( - 2 \beta - 1) q^{57} + ( - 6 \beta + 6) q^{58} + (2 \beta - 6) q^{59} - q^{60} + ( - 7 \beta + 8) q^{61} + (4 \beta - 2) q^{62} + ( - \beta + 2) q^{63} + q^{64} + ( - \beta + 1) q^{65} + \beta q^{66} + \beta q^{67} + ( - 3 \beta + 4) q^{68} + (2 \beta + 2) q^{69} + (\beta - 1) q^{70} + ( - 11 \beta + 9) q^{71} + ( - \beta + 2) q^{72} + 4 q^{73} - 4 q^{74} + (4 \beta + 1) q^{75} + (\beta + 1) q^{76} - q^{77} - \beta q^{78} + ( - 9 \beta + 3) q^{79} + (\beta - 1) q^{80} + ( - 6 \beta + 2) q^{81} + (8 \beta - 4) q^{82} + (5 \beta + 9) q^{83} + \beta q^{84} + (4 \beta - 7) q^{85} + ( - 5 \beta + 4) q^{86} - 6 q^{87} - q^{88} + ( - 3 \beta + 4) q^{89} + (2 \beta - 3) q^{90} + q^{91} - 2 \beta q^{92} + (2 \beta + 4) q^{93} + ( - 2 \beta - 4) q^{94} + \beta q^{95} + \beta q^{96} + ( - 4 \beta - 2) q^{97} - q^{98} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} + q^{10} + 2 q^{11} - q^{12} - 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 5 q^{17} + 3 q^{18} + 3 q^{19} - q^{20} + q^{21} - 2 q^{22} - 2 q^{23} + q^{24} - 7 q^{25} + 2 q^{26} + 2 q^{27} - 2 q^{28} - 6 q^{29} + 2 q^{30} - 2 q^{32} - q^{33} - 5 q^{34} + q^{35} - 3 q^{36} + 8 q^{37} - 3 q^{38} + q^{39} + q^{40} - q^{42} - 3 q^{43} + 2 q^{44} + 4 q^{45} + 2 q^{46} + 10 q^{47} - q^{48} + 2 q^{49} + 7 q^{50} + 5 q^{51} - 2 q^{52} + 17 q^{53} - 2 q^{54} - q^{55} + 2 q^{56} - 4 q^{57} + 6 q^{58} - 10 q^{59} - 2 q^{60} + 9 q^{61} + 3 q^{63} + 2 q^{64} + q^{65} + q^{66} + q^{67} + 5 q^{68} + 6 q^{69} - q^{70} + 7 q^{71} + 3 q^{72} + 8 q^{73} - 8 q^{74} + 6 q^{75} + 3 q^{76} - 2 q^{77} - q^{78} - 3 q^{79} - q^{80} - 2 q^{81} + 23 q^{83} + q^{84} - 10 q^{85} + 3 q^{86} - 12 q^{87} - 2 q^{88} + 5 q^{89} - 4 q^{90} + 2 q^{91} - 2 q^{92} + 10 q^{93} - 10 q^{94} + q^{95} + q^{96} - 8 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−1.00000 −1.61803 1.00000 0.618034 1.61803 −1.00000 −1.00000 −0.381966 −0.618034
1.2 −1.00000 0.618034 1.00000 −1.61803 −0.618034 −1.00000 −1.00000 −2.61803 1.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

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 2002.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2002.2.a.d 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2002))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$19$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 20 \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 80 \) Copy content Toggle raw display
$43$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$53$ \( T^{2} - 17T + 71 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$61$ \( T^{2} - 9T - 41 \) Copy content Toggle raw display
$67$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$71$ \( T^{2} - 7T - 139 \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 3T - 99 \) Copy content Toggle raw display
$83$ \( T^{2} - 23T + 101 \) Copy content Toggle raw display
$89$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
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