Properties

Label 4002.2.a.u
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} + \beta q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} + \beta q^{7} - q^{8} + q^{9} - \beta q^{10} + 2 \beta q^{11} + q^{12} + (2 \beta - 2) q^{13} - \beta q^{14} + \beta q^{15} + q^{16} - \beta q^{17} - q^{18} + ( - \beta - 4) q^{19} + \beta q^{20} + \beta q^{21} - 2 \beta q^{22} - q^{23} - q^{24} + (\beta - 1) q^{25} + ( - 2 \beta + 2) q^{26} + q^{27} + \beta q^{28} - q^{29} - \beta q^{30} - q^{32} + 2 \beta q^{33} + \beta q^{34} + (\beta + 4) q^{35} + q^{36} + (\beta + 6) q^{37} + (\beta + 4) q^{38} + (2 \beta - 2) q^{39} - \beta q^{40} + (\beta + 6) q^{41} - \beta q^{42} + (\beta + 4) q^{43} + 2 \beta q^{44} + \beta q^{45} + q^{46} + ( - 3 \beta + 4) q^{47} + q^{48} + (\beta - 3) q^{49} + ( - \beta + 1) q^{50} - \beta q^{51} + (2 \beta - 2) q^{52} + ( - 2 \beta + 4) q^{53} - q^{54} + (2 \beta + 8) q^{55} - \beta q^{56} + ( - \beta - 4) q^{57} + q^{58} + (3 \beta - 4) q^{59} + \beta q^{60} - 2 q^{61} + \beta q^{63} + q^{64} + 8 q^{65} - 2 \beta q^{66} + (2 \beta + 2) q^{67} - \beta q^{68} - q^{69} + ( - \beta - 4) q^{70} + 8 q^{71} - q^{72} + ( - 4 \beta - 2) q^{73} + ( - \beta - 6) q^{74} + (\beta - 1) q^{75} + ( - \beta - 4) q^{76} + (2 \beta + 8) q^{77} + ( - 2 \beta + 2) q^{78} + ( - 4 \beta + 6) q^{79} + \beta q^{80} + q^{81} + ( - \beta - 6) q^{82} + ( - 2 \beta + 2) q^{83} + \beta q^{84} + ( - \beta - 4) q^{85} + ( - \beta - 4) q^{86} - q^{87} - 2 \beta q^{88} + ( - 6 \beta + 8) q^{89} - \beta q^{90} + 8 q^{91} - q^{92} + (3 \beta - 4) q^{94} + ( - 5 \beta - 4) q^{95} - q^{96} - 8 q^{97} + ( - \beta + 3) q^{98} + 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - q^{14} + q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 9 q^{19} + q^{20} + q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} - q^{25} + 2 q^{26} + 2 q^{27} + q^{28} - 2 q^{29} - q^{30} - 2 q^{32} + 2 q^{33} + q^{34} + 9 q^{35} + 2 q^{36} + 13 q^{37} + 9 q^{38} - 2 q^{39} - q^{40} + 13 q^{41} - q^{42} + 9 q^{43} + 2 q^{44} + q^{45} + 2 q^{46} + 5 q^{47} + 2 q^{48} - 5 q^{49} + q^{50} - q^{51} - 2 q^{52} + 6 q^{53} - 2 q^{54} + 18 q^{55} - q^{56} - 9 q^{57} + 2 q^{58} - 5 q^{59} + q^{60} - 4 q^{61} + q^{63} + 2 q^{64} + 16 q^{65} - 2 q^{66} + 6 q^{67} - q^{68} - 2 q^{69} - 9 q^{70} + 16 q^{71} - 2 q^{72} - 8 q^{73} - 13 q^{74} - q^{75} - 9 q^{76} + 18 q^{77} + 2 q^{78} + 8 q^{79} + q^{80} + 2 q^{81} - 13 q^{82} + 2 q^{83} + q^{84} - 9 q^{85} - 9 q^{86} - 2 q^{87} - 2 q^{88} + 10 q^{89} - q^{90} + 16 q^{91} - 2 q^{92} - 5 q^{94} - 13 q^{95} - 2 q^{96} - 16 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
−1.00000 1.00000 1.00000 −1.56155 −1.00000 −1.56155 −1.00000 1.00000 1.56155
1.2 −1.00000 1.00000 1.00000 2.56155 −1.00000 2.56155 −1.00000 1.00000 −2.56155
\(n\): 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.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.u 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{2} - T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 16 \) 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
$5$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 16 \) 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
$37$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$41$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$43$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T - 128 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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