Properties

Label 4002.2.a.bg
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - \beta_{5} q^{5} + q^{6} + ( - \beta_{6} - \beta_{2} - 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - \beta_{5} q^{5} + q^{6} + ( - \beta_{6} - \beta_{2} - 1) q^{7} - q^{8} + q^{9} + \beta_{5} q^{10} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{11} - q^{12} + ( - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{13}+ \cdots + ( - \beta_{6} - \beta_{4} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu^{6} - 7\nu^{5} - 47\nu^{4} + 100\nu^{3} + 4\nu^{2} - 44\nu - 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -5\nu^{6} + 8\nu^{5} + 60\nu^{4} - 116\nu^{3} - 23\nu^{2} + 54\nu + 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 7\nu^{6} - 10\nu^{5} - 86\nu^{4} + 149\nu^{3} + 61\nu^{2} - 79\nu - 29 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -7\nu^{6} + 11\nu^{5} + 85\nu^{4} - 161\nu^{3} - 45\nu^{2} + 87\nu + 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 8\nu^{6} - 12\nu^{5} - 98\nu^{4} + 177\nu^{3} + 64\nu^{2} - 95\nu - 35 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -11\nu^{6} + 17\nu^{5} + 134\nu^{4} - 249\nu^{3} - 76\nu^{2} + 130\nu + 41 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + 3\beta_{5} - \beta_{3} - \beta_{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{5} + 5\beta_{4} + 2\beta_{3} + 11\beta_{2} + 9\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24\beta_{6} + 26\beta_{5} - 7\beta_{4} - 9\beta_{3} - 18\beta_{2} - 5\beta _1 + 69 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{6} + 30\beta_{5} + 47\beta_{4} + 33\beta_{3} + 122\beta_{2} + 95\beta _1 - 25 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 266\beta_{6} + 241\beta_{5} - 114\beta_{4} - 97\beta_{3} - 261\beta_{2} - 106\beta _1 + 698 ) / 2 \) 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
2.99277
−3.51756
2.02256
−0.518095
0.932053
−0.571135
−0.340585
−1.00000 −1.00000 1.00000 −3.83000 1.00000 −4.91072 −1.00000 1.00000 3.83000
1.2 −1.00000 −1.00000 1.00000 −0.694170 1.00000 1.42319 −1.00000 1.00000 0.694170
1.3 −1.00000 −1.00000 1.00000 −0.662522 1.00000 −0.537195 −1.00000 1.00000 0.662522
1.4 −1.00000 −1.00000 1.00000 −0.324686 1.00000 2.40824 −1.00000 1.00000 0.324686
1.5 −1.00000 −1.00000 1.00000 1.78524 1.00000 0.541489 −1.00000 1.00000 −1.78524
1.6 −1.00000 −1.00000 1.00000 2.26164 1.00000 −5.18548 −1.00000 1.00000 −2.26164
1.7 −1.00000 −1.00000 1.00000 3.46450 1.00000 1.26047 −1.00000 1.00000 −3.46450
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.bg 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bg 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{7} - 2T_{5}^{6} - 16T_{5}^{5} + 34T_{5}^{4} + 29T_{5}^{3} - 42T_{5}^{2} - 40T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{7} + 5T_{7}^{6} - 18T_{7}^{5} - 52T_{7}^{4} + 172T_{7}^{3} - 96T_{7}^{2} - 48T_{7} + 32 \) Copy content Toggle raw display
\( T_{11}^{7} + T_{11}^{6} - 37T_{11}^{5} - 47T_{11}^{4} + 248T_{11}^{3} + 204T_{11}^{2} - 456T_{11} - 176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( (T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 2 T^{6} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{7} + 5 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( T^{7} + T^{6} + \cdots - 176 \) Copy content Toggle raw display
$13$ \( T^{7} - T^{6} + \cdots - 6016 \) Copy content Toggle raw display
$17$ \( T^{7} + T^{6} + \cdots - 256 \) Copy content Toggle raw display
$19$ \( T^{7} + 9 T^{6} + \cdots - 1312 \) Copy content Toggle raw display
$23$ \( (T - 1)^{7} \) Copy content Toggle raw display
$29$ \( (T - 1)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} + 19 T^{6} + \cdots + 589312 \) Copy content Toggle raw display
$37$ \( T^{7} + 18 T^{6} + \cdots + 3112 \) Copy content Toggle raw display
$41$ \( T^{7} - 4 T^{6} + \cdots + 134408 \) Copy content Toggle raw display
$43$ \( T^{7} + 9 T^{6} + \cdots - 1364288 \) Copy content Toggle raw display
$47$ \( T^{7} + 11 T^{6} + \cdots + 88832 \) Copy content Toggle raw display
$53$ \( T^{7} - 8 T^{6} + \cdots - 536128 \) Copy content Toggle raw display
$59$ \( T^{7} - 12 T^{6} + \cdots - 95296 \) Copy content Toggle raw display
$61$ \( T^{7} + 5 T^{6} + \cdots - 22768 \) Copy content Toggle raw display
$67$ \( T^{7} + 13 T^{6} + \cdots + 203776 \) Copy content Toggle raw display
$71$ \( T^{7} + T^{6} + \cdots + 183296 \) Copy content Toggle raw display
$73$ \( T^{7} + 26 T^{6} + \cdots - 836032 \) Copy content Toggle raw display
$79$ \( T^{7} + 38 T^{6} + \cdots + 86848 \) Copy content Toggle raw display
$83$ \( T^{7} + 6 T^{6} + \cdots - 2048 \) Copy content Toggle raw display
$89$ \( T^{7} - 20 T^{6} + \cdots + 512 \) Copy content Toggle raw display
$97$ \( T^{7} + 8 T^{6} + \cdots - 407552 \) Copy content Toggle raw display
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