Properties

Label 4006.2.a.g
Level $4006$
Weight $2$
Character orbit 4006.a
Self dual yes
Analytic conductor $31.988$
Analytic rank $1$
Dimension $40$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4006,2,Mod(1,4006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4006.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: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −1.00000 −3.34571 1.00000 −3.64453 3.34571 5.09237 −1.00000 8.19376 3.64453
1.2 −1.00000 −3.11880 1.00000 1.57138 3.11880 2.57640 −1.00000 6.72691 −1.57138
1.3 −1.00000 −2.96542 1.00000 0.539090 2.96542 −0.220082 −1.00000 5.79371 −0.539090
1.4 −1.00000 −2.86035 1.00000 −2.90727 2.86035 0.924767 −1.00000 5.18162 2.90727
1.5 −1.00000 −2.39522 1.00000 0.491091 2.39522 3.07111 −1.00000 2.73707 −0.491091
1.6 −1.00000 −2.30611 1.00000 3.88128 2.30611 0.245423 −1.00000 2.31816 −3.88128
1.7 −1.00000 −2.23215 1.00000 −2.75227 2.23215 −2.11016 −1.00000 1.98248 2.75227
1.8 −1.00000 −1.94007 1.00000 −0.275091 1.94007 −2.80268 −1.00000 0.763873 0.275091
1.9 −1.00000 −1.91121 1.00000 −1.39712 1.91121 3.46874 −1.00000 0.652717 1.39712
1.10 −1.00000 −1.84543 1.00000 2.20410 1.84543 −0.412853 −1.00000 0.405623 −2.20410
1.11 −1.00000 −1.82093 1.00000 −3.80048 1.82093 −3.54475 −1.00000 0.315788 3.80048
1.12 −1.00000 −1.69686 1.00000 3.10338 1.69686 1.10094 −1.00000 −0.120661 −3.10338
1.13 −1.00000 −1.01040 1.00000 −1.91784 1.01040 4.92355 −1.00000 −1.97910 1.91784
1.14 −1.00000 −0.956879 1.00000 −0.0953014 0.956879 −4.20577 −1.00000 −2.08438 0.0953014
1.15 −1.00000 −0.941592 1.00000 −0.129713 0.941592 0.350515 −1.00000 −2.11340 0.129713
1.16 −1.00000 −0.934245 1.00000 −4.17073 0.934245 1.30479 −1.00000 −2.12719 4.17073
1.17 −1.00000 −0.779235 1.00000 −0.867411 0.779235 −3.45139 −1.00000 −2.39279 0.867411
1.18 −1.00000 −0.712075 1.00000 −4.09209 0.712075 2.33011 −1.00000 −2.49295 4.09209
1.19 −1.00000 −0.535052 1.00000 2.84024 0.535052 2.46812 −1.00000 −2.71372 −2.84024
1.20 −1.00000 −0.0866100 1.00000 2.47499 0.0866100 0.719708 −1.00000 −2.99250 −2.47499
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(2003\) \(1\)

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 4006.2.a.g 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4006.2.a.g 40 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + T_{3}^{39} - 74 T_{3}^{38} - 70 T_{3}^{37} + 2498 T_{3}^{36} + 2231 T_{3}^{35} - 51051 T_{3}^{34} - 42997 T_{3}^{33} + 706914 T_{3}^{32} + 561414 T_{3}^{31} - 7033157 T_{3}^{30} - 5274340 T_{3}^{29} + \cdots - 6140 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\). Copy content Toggle raw display