Properties

Label 4002.2.a.w
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: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} + q^{8} + q^{9} + 3 q^{10} + ( - \beta - 1) q^{11} + q^{12} + ( - \beta + 3) q^{13} + 3 q^{14} + 3 q^{15} + q^{16} + ( - 2 \beta - 3) q^{17} + q^{18} + (2 \beta - 1) q^{19} + 3 q^{20} + 3 q^{21} + ( - \beta - 1) q^{22} + q^{23} + q^{24} + 4 q^{25} + ( - \beta + 3) q^{26} + q^{27} + 3 q^{28} - q^{29} + 3 q^{30} + (\beta - 1) q^{31} + q^{32} + ( - \beta - 1) q^{33} + ( - 2 \beta - 3) q^{34} + 9 q^{35} + q^{36} + ( - 3 \beta - 2) q^{37} + (2 \beta - 1) q^{38} + ( - \beta + 3) q^{39} + 3 q^{40} + 7 \beta q^{41} + 3 q^{42} + (4 \beta - 5) q^{43} + ( - \beta - 1) q^{44} + 3 q^{45} + q^{46} + (6 \beta + 1) q^{47} + q^{48} + 2 q^{49} + 4 q^{50} + ( - 2 \beta - 3) q^{51} + ( - \beta + 3) q^{52} + (2 \beta + 6) q^{53} + q^{54} + ( - 3 \beta - 3) q^{55} + 3 q^{56} + (2 \beta - 1) q^{57} - q^{58} + ( - 2 \beta + 7) q^{59} + 3 q^{60} + (6 \beta - 4) q^{61} + (\beta - 1) q^{62} + 3 q^{63} + q^{64} + ( - 3 \beta + 9) q^{65} + ( - \beta - 1) q^{66} + ( - 2 \beta - 6) q^{67} + ( - 2 \beta - 3) q^{68} + q^{69} + 9 q^{70} + (5 \beta - 3) q^{71} + q^{72} + ( - 4 \beta - 6) q^{73} + ( - 3 \beta - 2) q^{74} + 4 q^{75} + (2 \beta - 1) q^{76} + ( - 3 \beta - 3) q^{77} + ( - \beta + 3) q^{78} + ( - 7 \beta - 5) q^{79} + 3 q^{80} + q^{81} + 7 \beta q^{82} + (2 \beta - 8) q^{83} + 3 q^{84} + ( - 6 \beta - 9) q^{85} + (4 \beta - 5) q^{86} - q^{87} + ( - \beta - 1) q^{88} + ( - 2 \beta - 2) q^{89} + 3 q^{90} + ( - 3 \beta + 9) q^{91} + q^{92} + (\beta - 1) q^{93} + (6 \beta + 1) q^{94} + (6 \beta - 3) q^{95} + q^{96} + 4 q^{97} + 2 q^{98} + ( - \beta - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} + 6 q^{14} + 6 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} + 6 q^{20} + 6 q^{21} - 2 q^{22} + 2 q^{23} + 2 q^{24} + 8 q^{25} + 6 q^{26} + 2 q^{27} + 6 q^{28} - 2 q^{29} + 6 q^{30} - 2 q^{31} + 2 q^{32} - 2 q^{33} - 6 q^{34} + 18 q^{35} + 2 q^{36} - 4 q^{37} - 2 q^{38} + 6 q^{39} + 6 q^{40} + 6 q^{42} - 10 q^{43} - 2 q^{44} + 6 q^{45} + 2 q^{46} + 2 q^{47} + 2 q^{48} + 4 q^{49} + 8 q^{50} - 6 q^{51} + 6 q^{52} + 12 q^{53} + 2 q^{54} - 6 q^{55} + 6 q^{56} - 2 q^{57} - 2 q^{58} + 14 q^{59} + 6 q^{60} - 8 q^{61} - 2 q^{62} + 6 q^{63} + 2 q^{64} + 18 q^{65} - 2 q^{66} - 12 q^{67} - 6 q^{68} + 2 q^{69} + 18 q^{70} - 6 q^{71} + 2 q^{72} - 12 q^{73} - 4 q^{74} + 8 q^{75} - 2 q^{76} - 6 q^{77} + 6 q^{78} - 10 q^{79} + 6 q^{80} + 2 q^{81} - 16 q^{83} + 6 q^{84} - 18 q^{85} - 10 q^{86} - 2 q^{87} - 2 q^{88} - 4 q^{89} + 6 q^{90} + 18 q^{91} + 2 q^{92} - 2 q^{93} + 2 q^{94} - 6 q^{95} + 2 q^{96} + 8 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 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
1.73205
−1.73205
1.00000 1.00000 1.00000 3.00000 1.00000 3.00000 1.00000 1.00000 3.00000
1.2 1.00000 1.00000 1.00000 3.00000 1.00000 3.00000 1.00000 1.00000 3.00000
\(n\): 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.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.w 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 2 \) 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
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T - 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 3 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$41$ \( T^{2} - 147 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T - 23 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 107 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 37 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 66 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$79$ \( T^{2} + 10T - 122 \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 52 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$97$ \( (T - 4)^{2} \) Copy content Toggle raw display
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