Properties

Label 4002.2.a.s
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} + q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} + q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9} + (\beta + 1) q^{10} - 2 q^{11} - q^{12} - 2 \beta q^{13} + ( - \beta - 1) q^{14} + (\beta + 1) q^{15} + q^{16} + (\beta - 5) q^{17} - q^{18} + ( - \beta + 1) q^{19} + ( - \beta - 1) q^{20} + ( - \beta - 1) q^{21} + 2 q^{22} - q^{23} + q^{24} + 3 \beta q^{25} + 2 \beta q^{26} - q^{27} + (\beta + 1) q^{28} + q^{29} + ( - \beta - 1) q^{30} - q^{32} + 2 q^{33} + ( - \beta + 5) q^{34} + ( - 3 \beta - 5) q^{35} + q^{36} + ( - \beta + 1) q^{37} + (\beta - 1) q^{38} + 2 \beta q^{39} + (\beta + 1) q^{40} + ( - \beta - 1) q^{41} + (\beta + 1) q^{42} + ( - 3 \beta + 7) q^{43} - 2 q^{44} + ( - \beta - 1) q^{45} + q^{46} + ( - \beta + 5) q^{47} - q^{48} + (3 \beta - 2) q^{49} - 3 \beta q^{50} + ( - \beta + 5) q^{51} - 2 \beta q^{52} - 2 q^{53} + q^{54} + (2 \beta + 2) q^{55} + ( - \beta - 1) q^{56} + (\beta - 1) q^{57} - q^{58} + ( - \beta + 1) q^{59} + (\beta + 1) q^{60} + ( - 2 \beta + 2) q^{61} + (\beta + 1) q^{63} + q^{64} + (4 \beta + 8) q^{65} - 2 q^{66} + (\beta - 5) q^{68} + q^{69} + (3 \beta + 5) q^{70} - 8 q^{71} - q^{72} + 6 q^{73} + (\beta - 1) q^{74} - 3 \beta q^{75} + ( - \beta + 1) q^{76} + ( - 2 \beta - 2) q^{77} - 2 \beta q^{78} + (4 \beta - 10) q^{79} + ( - \beta - 1) q^{80} + q^{81} + (\beta + 1) q^{82} + (2 \beta - 8) q^{83} + ( - \beta - 1) q^{84} + (3 \beta + 1) q^{85} + (3 \beta - 7) q^{86} - q^{87} + 2 q^{88} + ( - 2 \beta + 2) q^{89} + (\beta + 1) q^{90} + ( - 4 \beta - 8) q^{91} - q^{92} + (\beta - 5) q^{94} + (\beta + 3) q^{95} + q^{96} + (2 \beta + 8) q^{97} + ( - 3 \beta + 2) q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} - 9 q^{17} - 2 q^{18} + q^{19} - 3 q^{20} - 3 q^{21} + 4 q^{22} - 2 q^{23} + 2 q^{24} + 3 q^{25} + 2 q^{26} - 2 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 2 q^{32} + 4 q^{33} + 9 q^{34} - 13 q^{35} + 2 q^{36} + q^{37} - q^{38} + 2 q^{39} + 3 q^{40} - 3 q^{41} + 3 q^{42} + 11 q^{43} - 4 q^{44} - 3 q^{45} + 2 q^{46} + 9 q^{47} - 2 q^{48} - q^{49} - 3 q^{50} + 9 q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 6 q^{55} - 3 q^{56} - q^{57} - 2 q^{58} + q^{59} + 3 q^{60} + 2 q^{61} + 3 q^{63} + 2 q^{64} + 20 q^{65} - 4 q^{66} - 9 q^{68} + 2 q^{69} + 13 q^{70} - 16 q^{71} - 2 q^{72} + 12 q^{73} - q^{74} - 3 q^{75} + q^{76} - 6 q^{77} - 2 q^{78} - 16 q^{79} - 3 q^{80} + 2 q^{81} + 3 q^{82} - 14 q^{83} - 3 q^{84} + 5 q^{85} - 11 q^{86} - 2 q^{87} + 4 q^{88} + 2 q^{89} + 3 q^{90} - 20 q^{91} - 2 q^{92} - 9 q^{94} + 7 q^{95} + 2 q^{96} + 18 q^{97} + q^{98} - 4 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−1.00000 −1.00000 1.00000 −3.56155 1.00000 3.56155 −1.00000 1.00000 3.56155
1.2 −1.00000 −1.00000 1.00000 0.561553 1.00000 −0.561553 −1.00000 1.00000 −0.561553
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(23\) \( +1 \)
\(29\) \( -1 \)

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 4002.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} - 11T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 16T - 4 \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
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