Properties

Label 4002.2.a.be
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.2389280.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 11x^{3} + 7x^{2} + 26x - 18 \) 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta_{2} q^{5} + q^{6} + (\beta_{4} - 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + \beta_{2} q^{5} + q^{6} + (\beta_{4} - 1) q^{7} - q^{8} + q^{9} - \beta_{2} q^{10} + ( - \beta_{2} + \beta_1 + 1) q^{11} - q^{12} + ( - \beta_1 - 2) q^{13} + ( - \beta_{4} + 1) q^{14} - \beta_{2} q^{15} + q^{16} + (\beta_{4} - 2 \beta_1 + 1) q^{17} - q^{18} + ( - \beta_{4} + 2 \beta_1 - 1) q^{19} + \beta_{2} q^{20} + ( - \beta_{4} + 1) q^{21} + (\beta_{2} - \beta_1 - 1) q^{22} - q^{23} + q^{24} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{25} + (\beta_1 + 2) q^{26} - q^{27} + (\beta_{4} - 1) q^{28} - q^{29} + \beta_{2} q^{30} + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{31}+ \cdots + ( - \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 5 q^{12} - 11 q^{13} + 4 q^{14} - q^{15} + 5 q^{16} + 4 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 5 q^{22} - 5 q^{23} + 5 q^{24} + 12 q^{25} + 11 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 5 q^{33} - 4 q^{34} - 6 q^{35} + 5 q^{36} + 9 q^{37} + 4 q^{38} + 11 q^{39} - q^{40} + 5 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + q^{45} + 5 q^{46} + 8 q^{47} - 5 q^{48} - 5 q^{49} - 12 q^{50} - 4 q^{51} - 11 q^{52} - 4 q^{53} + 5 q^{54} - 33 q^{55} + 4 q^{56} + 4 q^{57} + 5 q^{58} + 23 q^{59} - q^{60} + 7 q^{61} - 5 q^{62} - 4 q^{63} + 5 q^{64} - 5 q^{65} + 5 q^{66} + 5 q^{67} + 4 q^{68} + 5 q^{69} + 6 q^{70} - 21 q^{71} - 5 q^{72} - 8 q^{73} - 9 q^{74} - 12 q^{75} - 4 q^{76} - 11 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 5 q^{82} - 18 q^{83} + 4 q^{84} - 10 q^{85} + 16 q^{86} + 5 q^{87} - 5 q^{88} + 32 q^{89} - q^{90} + 10 q^{91} - 5 q^{92} - 5 q^{93} - 8 q^{94} + 10 q^{95} + 5 q^{96} - 16 q^{97} + 5 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 11x^{3} + 7x^{2} + 26x - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 8\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 8\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{3} - 2\nu^{2} - 6\nu + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + 2\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} - 9\beta_{3} + 11\beta_{2} + 4\beta _1 + 24 \) 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
1.61972
−2.11043
0.711151
−2.35912
3.13869
−1.00000 −1.00000 1.00000 −2.88781 1.00000 −3.71597 −1.00000 1.00000 2.88781
1.2 −1.00000 −1.00000 1.00000 −2.25808 1.00000 1.35501 −1.00000 1.00000 2.25808
1.3 −1.00000 −1.00000 1.00000 −0.651506 1.00000 2.08128 −1.00000 1.00000 0.651506
1.4 −1.00000 −1.00000 1.00000 3.26670 1.00000 −3.10583 −1.00000 1.00000 −3.26670
1.5 −1.00000 −1.00000 1.00000 3.53070 1.00000 −0.614480 −1.00000 1.00000 −3.53070
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.be 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.be 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{5} - T_{5}^{4} - 18T_{5}^{3} + 4T_{5}^{2} + 85T_{5} + 49 \) Copy content Toggle raw display
\( T_{7}^{5} + 4T_{7}^{4} - 7T_{7}^{3} - 26T_{7}^{2} + 20T_{7} + 20 \) Copy content Toggle raw display
\( T_{11}^{5} - 5T_{11}^{4} - 17T_{11}^{3} + 99T_{11}^{2} - 122T_{11} + 42 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - T^{4} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{5} + 4 T^{4} + \cdots + 20 \) Copy content Toggle raw display
$11$ \( T^{5} - 5 T^{4} + \cdots + 42 \) Copy content Toggle raw display
$13$ \( T^{5} + 11 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} + \cdots - 1764 \) Copy content Toggle raw display
$19$ \( T^{5} + 4 T^{4} + \cdots + 1764 \) Copy content Toggle raw display
$23$ \( (T + 1)^{5} \) Copy content Toggle raw display
$29$ \( (T + 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 5 T^{4} + \cdots - 274 \) Copy content Toggle raw display
$37$ \( T^{5} - 9 T^{4} + \cdots + 1665 \) Copy content Toggle raw display
$41$ \( T^{5} - 5 T^{4} + \cdots - 815 \) Copy content Toggle raw display
$43$ \( T^{5} + 16 T^{4} + \cdots - 168 \) Copy content Toggle raw display
$47$ \( T^{5} - 8 T^{4} + \cdots - 35084 \) Copy content Toggle raw display
$53$ \( T^{5} + 4 T^{4} + \cdots - 4480 \) Copy content Toggle raw display
$59$ \( T^{5} - 23 T^{4} + \cdots - 3507 \) Copy content Toggle raw display
$61$ \( T^{5} - 7 T^{4} + \cdots + 46708 \) Copy content Toggle raw display
$67$ \( T^{5} - 5 T^{4} + \cdots - 23448 \) Copy content Toggle raw display
$71$ \( T^{5} + 21 T^{4} + \cdots - 1098 \) Copy content Toggle raw display
$73$ \( T^{5} + 8 T^{4} + \cdots + 152368 \) Copy content Toggle raw display
$79$ \( T^{5} + 8 T^{4} + \cdots - 24 \) Copy content Toggle raw display
$83$ \( T^{5} + 18 T^{4} + \cdots + 412576 \) Copy content Toggle raw display
$89$ \( T^{5} - 32 T^{4} + \cdots - 8640 \) Copy content Toggle raw display
$97$ \( T^{5} + 16 T^{4} + \cdots + 13632 \) Copy content Toggle raw display
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